THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 

GIFT  OF 

John  S.Preil 


• 


\ 


PRINCIPLES 


OP 


THE  MECHANICS 


OF 


MACHINERY   AND   ENGINEERING. 


BY  JULIUS  WEISBACH, 

PROFESSOR  OF  MECHANICS  AND  APPLIED-MATHEMATICS  IN  THE  ROYAL  MINING 
ACADEMY  OF  FREIBERG. 

FIRST  AMERICAN  EDITION. 

EDITED 
BY  WALTER  R.  JOHNSON,  A.M.,  CIV.  &  MIN.  ENG. 

WASHINGTON,    B.C. 

FORMERLY  PROFESSOR  OF  MECHANICS  AND  NATURAL  PHILOSOPHY  IN  THE  FRANKLIN  INSTI- 
TUTE, AND  OF  CHEMISTRY  AND  NATURAL  PHILOSOPHY  IN  THE  MEDICAL 
DEPARTMENT  OF  PENNSYLVANIA  COLLEGE.       AUTHOR  OF  A 
REPORT  TO  THE  UNITED  STATES   NAVY  DEPART- 
MENT ON  AMERICAN  COALS,  &C.  &C. 


IN  TWO  VOLUMES. 

ILLUSTRATED  WITH  EIGHT  HUNDRED  AND  THIRTEEN 
ENGRAVINGS  ON  WC 


VOL.  II. 

APPLIED    ME 


PHILADELPHIA: 
LEA    AND    BLANCHARD. 

1849. 


ENTERED  according  to  the  Act  of  Congress,  in  the  year  1849,  by 

LEA  AND  BLANCHARD, 
in  the  Clerk's  Office  of  the  District  Court  of  the  Eastern  District  of  Pennsylvania. 


PHILADELPHIA: 
T.  K.  AND  P.  G.  COLLINS.  PRINTERS. 


Engmeeriflg 
Library 

TA 


PREFACE  BY  THE  AMERICAN  EDITOR. 


IN  submitting  to  the  American  reader  the  second  volume  of 
Weisbach's  Mechanics  of  Machinery  and  Engineering,  we  cannot, 
perhaps,  better  express  our  own  appreciation  of  the  value  of  this 
part  of  his  labors,  than  %  citing  a  passage  from  the  advertisement 
of  the  English  translator,  Prof.  L.  Gordon. 

"  The  usefulness  of  this  second  volume  will  be  manifest  from  the 
practical  interest  and  importance  of  the  subjects  treated.  The  first 
part  of  the  volume,  though  far  from  giving  a  complete  theory  of 
engineering  and  architectural  construction,  brings  many  important 
questions  of  practice  before  the  student  in  a  simple  form,  and  in  a 
light  by  which  he  will  more  readily  recognize  the  bearings  of  the 
mathematical  calculations  on  this  subject,  than  has  been  usually  the 
case  in  English  works.  The  second  part  of  the  volume  contains 
the  only  Theoretical  Treatise  on  Water  Power  of  the  least  practical 
value  hitherto  printed  in  the  English  language.  The  real  import- 
ance of  such  a  treatise  will  be  variously  estimated ;  but  as  it  is  the 
first  publication  in  which  a  systematic  attempt  is  made  to  familiarize 
English  Machinists  with  the  application  of  exact  reasoning  in  deve- 
loping the  theory  of  the  machines  treated  of,  it  is  believed  that  it 
must  be  interesting  to  them,  and  if  so,  it  cannot  fail  to  be  useful 
likewise." 

The  most  available  treatise  on  the  numerous  forms  of  reaction 
wheels,  and  other  turbines  to  which  the  American  student  has  access, 
is  believed  to  be  embraced  in  this  volume.  The  author,  it  may  be 
observed,  has  not  contented  himself  with  giving  a  general  theory  on 
that  subject,  but  by  skillfully  analyzing  the  several  effects  produced, 
and  computing  separately  the  prejudicial  and  the  useful  resistances 

713797 

""-  Engineering 

Library 


vi  PREFACE  BY  THE  AMERICAN  EDITOR. 

to  the  action  of  the  water,  has  presented  conclusions  challenging  the 
highest  confidence,  especially  as  they  stand  confirmed,  in  most  cases, 
by  the  results  of  numerous  direct  experiments. 

In  reference  to  the  water-pressure  engine,  also,  it  may  be  said  that 
the  present  volume  will  afford  to  the  American  student  the  most 
direct  and  positive  information  as  to  the  useful  application  of  water 
in  that  species  of  motor. 

In  the  original  work  of  Prof.  Weisbach,  the  second  volume  em- 
braced the  science  applicable  to  the  steam  engine,  but  as  that  sub- 
ject has  now  assumed  so  distinct  an  importance,  and  as  its  numerous 
topics  and  improvements  could  scarcely  be  presented  with  sufficient 
clearness,  in  a  less  space  than  an  entire  volume,  it  has  been  deemed 
expedient,  in  imitation  of  the  English  translator,  to  reserve  that 
branch  of  the  mechanics  of  engineering  for  a  separate  treatise. 

In  assigning  to  their  appropriate  chapters  the  additions  of  the 
translator,  which  had  in  the  English  edition  been  thrown  into  the 
form  of  an  appendix,  we  have  been  guided  by  a  desire  of  rendering 
the  work  more  serviceable  to  the  student,  by  placing  before  him  the 
whole  matter  pertaining  to  each  branch  under  its  appropriate  head. 
We  have  added  a  few  articles  particularly  relating  to  the  strength 
of  materials,  which,  we  hope,  may  not  be  found  uninteresting  to  the 
student.  Indeed,  when  we  take  into  view  the  lamentable,  and  often 
wilful  and  obstinate  disregard  of  the  truths  which  science  has  elicited 
relative  to  this  department  of  our  subject ;  when  we  see  machines 
and  engines  intended  to  perform  the  most  powerful  operations,  and 
edifices,  or  monuments,  designed  to  endure  for  ages,  constructed  of 
materials,  either  utterly  worthless,  or,  at  best,  of  very  inferior  cha- 
racter and  durability,  or  containing  in  their  composition  the  elements 
of  weakness  and  decay,  we  may  estimate,  with  some  justness,  the 
importance  of  those  researches  and  computations,  which  prove  what 
may  be  expected  from  the  employment  of  good  or  bad  materials 
respectively,  for  any  of  the  purposes  of  the  architect  and  engineer. 
The  fact  that  the  public  has  often  been  basely  imposed  upon  by 
reason  of  employing  as  architects  and  engineers  those  who  would 
pander  to  the  cupidity  of  contractors  for  materials  and  labor,  and 


PKEFACE  BY  THE  AMERICAN  EDITOR.  vil 

erect  public  works  wholly  discreditable  to  the  nation,  is  an  additional 
reason  why  works,  written  for  the  purpose  of  imparting  correct 
information  on  the  physical  properties  and  the  relative  values  of 
materials,  ought  to  be  diligently  studied  by  those  who  desire  correct 
and  reliable  knowledge. 

The  list  of  illustrations  which  we  have  added  will  much  facilitate 
reference  to  the  several  topics  to  which  they  relate,  and  the  execu- 
tion of  the  cuts,  with  the  creditable  manner  in  which  they  have  been 
used  by  the  printer,  will  be  sufficiently  apparent  to  the  most  casual 
observer. 

WASHINGTON,  August,  1849. 


AUTHOR'S   PREFACE. 


IN  writing  this,  the  second  volume,  I  have  adhered  as  closely  as 
possible  to  my  views  of  what  the  work  should  be,  as  explained  in 
the  preface  to  the  first  volume. 

I  am  aware  that  these  views  are  not  adopted  by  all  who  are 
capable  of  judging  in  the  matter,  and  that  a  more  general  and 
mathematical  treatment  of  the  subject  would  have  been  preferred  by 
many.  But  I  have  now  long  experience  in  teaching  to  fall  back 
upon,  and  am  thereby  convinced,  that  the  comparatively  elementary 
style  adopted  as  it  can  be  followed  by  those  who  have  not  made 
extensive  mathematical  acquirements,  will  more  surely  lead  to  the 
introduction  of  applications  of  Mechanical  Science  in  the  routine 
practice  of  engineers,  than  the  more  general  methods  of  treating 
these  subjects  have  done. 

A  basis  on  true  principles  and  established  facts,  and  simplicity  in 
the  method  of  analysis,  are  the  main  requisites  in  a  work  intended 
for  the  instruction  and  guidance  of  practical  men.  And  it  is  chiefly 
the  want  of  these,  in  technical  literature,  that  has  retarded  the 
introduction  of  science  amongst  those  engaged  in  the  execution  of 
works,  and  the  erection  of  machinery.  If  in  evolving  rules  of  art, 
imperfect  facts  be  assumed,  or  unwarranted  hypotheses  be  adopted — 
if  the  essential  be  not  distinguished  from  that  which  is  merely  col- 
lateral, and  if  important  considerations  be  neglected,  it  cannot  be 
expected  that  the  rules  deduced,  however  correct  the  process  of 
deduction,  will  be  available  for  any  useful  application.  But  this  is 
no  uncommon  fault.  Authors  forget  that  the  mathematics  can  only 
guide  our  ideas,  and  not  give  us  any :  and  thus,  in  admiration  of 
their  analytical  processes,  they  often  overlook  the  worthlessness  of 
the  premises.  Hence  it  arises  that  practicians  not  unfrequently 
reproach  theory  as  valueless,  whilst  it  is,  in  reality,  the  facts  of  the 
case  that  have  been  erroneously  stated  or  applied.  Besides,  it  is 
not  an  easy  matter  to  deduce  rules  of  art  by  the  principles  of 


X  AUTHOR S  PREFACE. 

science ;  for  this  requires  not  only  an  intimate  acquaintance  with 
the  subject  investigated,  but  generally  requires  special  observations 
or  experiments  to  be  made,  in  order  to  create  the  facts,  so  to  speak, 
that  are  to  be  reasoned  upon  and  reduced  to  a  theory  which  shall 
interpret  them. 

In  this  second  volume  of  his  work,  the  Author  has  done  his  utmost 
to  develop  theories  that  will  be  found  applicable  in  practice — to 
furnish  the  guide  above  alluded  to — well  aware,  however,  that  his 
endeavors  have  only  imperfectly  succeeded. 

This  volume  is  divided  into  two  parts ;  the  first,  the  application 
of  Mechanics  in  Construction,  and  the  other  to  the  theory  of  Ma- 
chines recipients  of  Water  and  Wind  Power.  The  Author  regrets 
now  his  not  having  entered  more  at  large  into  a  discussion  of  the 
theory  of  the  construction  of  wooden  and  stone  bridges,  and  more 
particularly  not  to  have  been  able  to  avail  himself  of  $he  information 
contained  in  Ardant's  Etudes  sur  I' etablissement  des  charpentes  a 
grande  portee,  as  this  subject  is,  in  these  times  of  railway  extension, 
of  especial  importance  (in  Germany). 

The  second  part  of  the  volume  is  as  concisely  written  as  was  con- 
sistent with  the  object  I  had  in  view.  I  now  regret  having  been  so 
brief  on  the  important  subject  of  Dynamometers.  The  chapter  on 
Turbines  may  appear  to  some  to  err  in  excess,  from  my  having  given 
the  details  of  the  theory  and  construction  of  the  old  impact  and 
pressure  turbines ;  but  I  consider  that  it  is  important  to  be  aware  of 
the  faults  or  imperfections  of  one  construction  of  a  machine,  in 
order  fully  to  appreciate  the  improvements  introduced  in  a  more 
perfect  one.  Again,  the  application  of  Water-pressure  Engines, 
being  almost  entirely  confined  to  the  Mining  Engineer's  province, 
the  fullness  with  which  I  have  treated  this  engine  may  appear  to 
exceed  its  relative  importance.  The  circumstance,  however,  that 
there  is  no  work  in  any  language,  that  I  am  aware  of,  treating  of 
these  engines,  must  be  my  apology  for  attempting  to  fill  that  gap  in 
technical  literature. 

I  hope  soon  to  preface  a  volume,  containing  a  Treatise  on  Me- 
chanism, and  on  the  principle  Operators,  or  machines  performing 
various  mechanical  operations. 

JULIUS  WEISBACH. 

FREIBERG,  December,  1847. 


S.  PRELL 

Civil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 
TABLE   OP  CONTENTS. 


PREFACE  BY  THE  AMERICAN  EDITOR 
AUTHOR'S  PREFACE 


SECTION    I. 

THE  APPLICATION  OF  MECHANICS  IN  BUILDINGS. 

CHAPTER  I. 
The  equilibrium  and  pressure  of  semi-fluids 13 

CHAPTER  H. 
Theory  of  arches 27 

CHAPTER  in. 

Theory  of  framings  of  wood  and  iron        ......       40 

Strength  of  materials 68 


DIVISION    II. 

APPLICATION   OF  MECHANICS  TO   MACHINERY. 

Introduction 99 

Rigidity  of  cordage 102 


Xll  CONTENTS. 

PAGE 

SECTION   II. 

OP   MOVING  POWERS,   AND  THEIR  EFFECTS. 

CHAPTER  I. 

Of  the  measure  of  moving  powers,  and  their  efiects  .         .         .         .105 

CHAPTER  H. 

Of  animal  power,  and  its  recipient  machines     .         .         .         .         .121 

CHAPTER  m. 

On  collecting  and  leading  water  that  is  to  serve  a  power      .         .         .     135 

CHAPTER  IV. 
Of  vertical  water  wheels 164 

CHAPTER  V. 

Of  horizontal  water  wheels 228 

CHAPTER  VI. 
Water-pressure  engines 298 

CHAPTER  VH. 
On  windmills         ...  .     341 


LIST  OF  ILLUSTRATIONS. 


NO.  PAG  I 

1.  Angle  of  repose  of  disintegrated  masses    -  13 

2.  Theory  of  the  pressure  of  earth     -  -                                                   14 

3.  Surcharged  masses  of  earth             -  17 

4.  Retaining  walls       -                          .  -           18 

5.  Slipping  of  walls    ...  -           19 

6.  Resistance  of  earth              -  ....           20 

7.  Depth  of  foundations           -  21 

8.  9.  Heeling  of  retaining  walls          -  .           22 
10.  Retaining  walls  with  batter  .           26 
H,  12.  Dislocation  of  arches  by  slipping  of  voussoirs  -             -             .           28 

13.  Line  of  pressure  and  resistance      ...  -28 

14,  15,  16.  Dislocation  of  arches  by  rotation      -  .           29 
17,  18,  19,  20.  Equilibrium  in  reference  to  rotation  -                                                   30 

21.  Stability  of  abutments         -  -           31 

22,  23.  Loaded  arches                             -  -           33 

24.  Test  of  the  equilibrium  of  arches               -  -             ...            .           34 

25.  Semicircular  arches             -----  «             -           36 

26.  27,  28,  29.  30.  Theory  of  framings  of  wood  and  iron          -            -           40,  41,  42 
31,  32,  33.  Thrust  of  roofs        -            -  -            .            •             -     42, 43 

34.  Compound  roofs      ...  44 

35,  36.  Supported  rafters          -  .    45,  46 

37.  King  posts  -            -  -          46 

38,  39,  and  40.   Trusses             -  -    47, 48 
41, 42,  43,  44, 45,  and  46.  Timber  bridges        -  .           48, 49,  50 

47.  Roofs           ...  .                                    50 

48,  49,  50,  51,  52,  53,  54,  55,  56,  57,  and  58.  Posts  -                        .           50,  52,  53 
59,  60,  61.  Braces  or  struts         -  -                                       -54 
62,  63,  64,  65,  66.  Compound  beams     -  •                                          55,  56 

67.  Lattice-framed  beams          -            -  -                          -           56 

68,  69,  70.  Curved  beams          -  -                                       -    56,  57 
71,  72,  73,  74,  75,  76,  77,  78,  79.  Chain  or  suspension  bridges  59  to  62 
80,  81.  Piers  and  abutments     -             -  -           66  and  67 

82.  Tenacity  of  iron,  treated  by  thermotension  ....           74 

83.  Conway  tubular  bridge        -             -  .82 

84.  85,  86,  87,  88,  89.  Tubular  bridges,  detailed  construction  of  84,  85,  91,  92,  93 
90,  91.  Useful  and  prejudicial  resistance            -  101,  103 
92,  93,  94.  Common  balance     -  -106 
95,  96,  97,  98.  Unequal  armed  balances  •          110  to  111 
99,  100,  101.  Weigh  bridges     -                        -  -          112  to  113 

VOL.  II. — 1 


LIST  OP  ILLUSTRATIONS. 


102.  George's  weighing  table      - 

103,  104,  105,  106,  107.  Index  balances,  or  spring  balances         -  - 
108,  109,  110,  111.  Friction  brake          -  - 
112  Dolly 

113,  114.  Diagrams  of  Bouguer's  formula 

1J5,  116,  117.  Levers     -  • 

118,  119,  120,  121.  Windlass     -  -  - 

122.  Vertical  capstans    •  •  - 

123.  Horse  capstans 

124.  125.  Diagrams  explanatory  of  horse  gins    - 

126,  127,  128.  Tread  wheels       ...  .. 

129.  Movable  inclined  plane 

130.  132.  Construction  of  weirs  - 

131.  Overfall  weir 

133.  Manner  of  constructing  wooden  weirs 

134.  Sluice 

135.  136.  Height  of  swell 

137,  138.  Diagrams  of  back  water          - 
139,  140,  141,  142.  Discontinuous  weirs 

143.  Amplitude  of  the  back  water          -  • 

144,  145.  Backwater  swell          • 
146,  147.  Dams 

148.  Cross  section  of  dykes         -  - 

149.  Stability  of  dykes    - 

]  50.  Offlet  sluices  for  dykes         - 

151,  152.  Water  leads     -  - 

153,  154.  Water  tunnels  - 

155,  156.  Construction  of  water  tubes     ----- 

157.  Flood  gates  -  - 

158.  Syphon  for  waste  water      - 

159.  160.  Sluices, 

161,  162,  163,  164,  165,  166,  167.  Conduit  pipes 

168.  Ventilator  pipes       - 

169.  Waste  cock  -  • 

170.  Forked  pipes  -  - 

171.  172.  Overshot  water  wheels  - 

173,  174,  175,  176,  177,  178,  179,  180,  181,  182,  183.  Axles  and  gudgeons 

184.  Proportions  of  water  wheels  -  - 

185,  186,  187,  188.  Form  of  buckets       -  -  - 
189,  190,  191.  Line  of  pitch  of  buckets  - 

192,  193,  194,  195,  196,  197,  198,  199.     Sluices  -          177, 

200.  Effect  of  impact      - 

201,  202.  Effect  of  water's  height  ..... 
203,  204.  Number  of  buckets     -  -  .  -  - 

205,  206,  207.  Efiect  of  centrifugal  force 

208,  209.  Useful  effect  of  wheels  -  -  -  -  . 

210,  211,  212.  High  -breast  wheels          .  •  .'•*..  . 

213.  Breast  wheel  with  inside  sluice       -  .. 

214,  215,  2  10.  Overfall  sluices    ...... 

217,  218.  Penstock  sluices  -  .  .  -  -  - 


116  to  118 
119  to  m 

127>  12: 
129,131 

-  131 
1  *3  1 

. 

-  141 
142,  143 

-  144 

147,  148 

148,  149 
-150 
-150 
-152 

-  154 
-154 

-  155 
155 
155 

157,  158 

-  159 
160 
160 

-  162 

-  167 
170,  171,  172 

172 

174,  175 

176,  177 

178,  179,  180 

181 

.         183 

186,  187 

188,  189 

192 

-  196 
.         197 
.         199 

-  201 


LIST  OF  ILLUSTRATIONS. 


SO.  PAGE 

219, 220, 221, 222.  Construction  of  the  curb  •  -  -      202,203, 205 

223.  Diagram  of  losses  -  -         206 

224.  Efficiency  of  breast  wheels,  diagram  .....         210 
225, 226.  Undershot  wheels        -            -  -  -  -  211 

227.  Wheel  in  straight  course     -.          -             •            -            •  -213 

228.  Useful  eSect  of  undershot  wheel    -  214 

229.  Boat  mill     -  216 

230.  Poncelet's  wheels   -                          -                                       -  -218 

231, 232.  Theory  of  Poncelet's  wheels               -             -            -            -  219,221 

233.  Curved  water  course  .......         226 

234, 235.  Small  wheels                                          -  -         227 

236,  237,  238,  239.  Impact  wheels          -                                                    -  229,  230,  231 

240,  241.  Impact  and  reaction  wheels    .....  232,233 

242, 243.  Borda's  turbines           ....  .         234 

244.  Pit  wheel    ......  .  235 

245.  Burdin's  turbines     -  -----  -  236 

246.  Poncelet's  turbine    -  -  238 

247.  Danai'des  or  pear  wheel      ....  239 

248.  Diagram  representing  the  reaction  of  water  ....  240 

249.  Machine  to  measure  the  reaction  of  water               ....  241 
250, 251.  Diagrams  illustrating  reaction               .....  242 
252, 253, 254.  Reaction  wheels               ......  242 

255.  Whitelaw's  turbine  .....  .         245 

256.  Combe's  reaction  wheel      -  ...  .        246 

257.  Cadiat's  turbine       -  .....         247 

258.  Redtenbacher's  water-tight  joint     ......         247 

259.  260,  261,  262,  263,  264.  Fourneyron's  turbine         -  -  •         248  to  553 
265, 266.  Cadiat's  footstep  for  turbines   -                          ...  253, 254 

267.  Theory  of  reaction  turbines,  diagram          .....         255 

268.  Sluices  for  turbines  -  •  -         264 

269.  Guide-curves  of  Gallon's  turbine     -  ...        265 

270.  Sluice  for  impact  and  pressure  turbines       .....         266 

271.  272.  Diagrams  to  explain  construction  of  guide-curve  turbines       -  270,  272 

273.  Whitelaw's  turbines  .......         279 

274.  Comparison  of  turbines       .......         281 

275. 276.  Fontaine's  turbine        •  ...         284 

276. 277.  Jonval's  turbine  .  .  -  285, 286 

278.  Theory  of  Jonval's  and  Fontaine's  turbine  -            .            -286 

279.  Construction  of  the  buckets             -  ....         290 

280.  281,  282,  283.  Turbines  with  horizontal  axes  -             -            -             295,296 
284,  285,  286,  287.  Water-pressure  engines         -  -                           298, 299 
288, 289, 290.  Pressure  pipes      -                          -  300 
291.  Bend  or  knee  piece              ...  ...         302 

292, 293.  The  working  piston     -  .....         302 

294.  Compensation  joint              .....  -         303 

295.  Piston  rod  and  stuffing-box               ....  -         304 
296, 297, 298, 299.  Valves          ....  304,  305 
300,  301,  302.  Slide-piston  valves            -            -  306,  307 

303.  Valve  gear  ......  -         307 

304,  305.  Counter-balance  gear  .....  308,  309 


LIST  OF  ILLUSTRATIONS. 


306,  307,  308.  Auxiliary  water-engine  valve  gear 
309,  310.  The  valve  cylinders    - 

311.  Saxon  water-pressure  engine 

312.  Huelgoat  water-pressure  engine 

313.  314,  315.  Darlington's  water-pressure  engine 
316,  317.  Adjustment  of  the  valves 

318,  319,  320.  Chain  wheels      - 

321.  Postmill      - 

322,  323.  Smockmills     - 

324,  325.  Regulation  of  the  power 
326,  327,  328.  Anemometers      - 

329.  Diagram  representing  the  force  of  wind    - 

330,  331.  Best  angle  of  impulse 


PAGE 

311  to  313 
314 
316 
318 

319  to  321 

332,  335 

339,  340 

344 

345,  346 
347,  348 
351,  352 
353 
354,  356 


PRINCIPLES 


THE    MECHANICS 


MACHINERY    AND    ENGINEERING 


SECTION  I. 

THE  APPLICATION  OF  MECHANICS  IN  BUILDING. 


CHAPTER   I. 

OF   THE    EQUILIBRIUM   AND    PRESSURE    OF   SEMI-FLUIDS. 

§  1.  Sand,  earth,  corn  seeds,  shot,  $c.  £c.,  may  be  considered  as 
semi-fluids. — They  resemble  fluids  in  so  far  as,  like  these,  they  require 
external  support  that  they  may  preserve  a  particular  form.  The 
mutual  adhesion  of  the  parts  of  semi-fluids  is  of  course  greater  than 
in  the  case  of  water.  Water  always  requires  external  support, 
while  this  is  only  the  more  frequent  case  with  so  called  semi-fluids; 
and  whilst  water  is  in  equilibrium  only  when  its  surface  is  horizontal, 
the  disintegrated  masses  or  semi-fluids  in  question,  may  be  in  stable 
equilibrium,  though  their  surface  be  inclined. 

If  the  parts  of  a  disintegrated  mass  be  connected  by  their  mutual 
friction  alone,  the  mass  will  be  in  equilibrium 
when  its  surface  is  not  inclined  to  the  horizon  Fis-  *• 

at  a  greater  angle  than  the  angle  of  repose  p 
(Vol.  I.  §  159).  The  natural  slope  of  disinte- 
grated masses  is  determined  by  the  angle  of  re- 
pose. If  by  the  slope  of  a  declivity AB,  Fig.  1, 

we  understand  the  ratio  -  of  the  base  AC—b  to 

a 
VOL.  n. — 2 


14  PRESSURE  OF  EARTH. 

tlie  height  CB=a,  it  is  evidently =  cotang.  P,  or,  as  tang.  p  is  equal 
to  the  co-efficient  of  friction  /$  -  =  -. 

According  to  Martony  de  Koszegh,  the  natural  slope  of  perfectly 
dry  soil,  for  example,  =  1,243,  and  for  moist  soil=  1,083.  Hence, 
the  angle  of  slope  in  the  first  case  is=  39°,  and  in  the  second  43°. 

For  very  fine  sand,  the  slope  =  |,  therefore,  the  angle  of  slope 
=  31°.  For  rye  seeds,  the  author  found  P  =  30°,  for  fine  shot 
v  =  25°,  and  for  the  finest  shot  P  =  22  J°. 

Remark.  Experiments  on  the  slope  of  disintegrated  masses  are  made  by  heaping  them 
up,  and  dressing  them  off  from  below  upwards. 

§  2.  Pressure  of  Earth. — If  a  disintegrated  mass,  such  as  earth, 
be  supported  by  a  retaining  wall,  it  exerts  a  pressure  (poussee) 
against  it,  a  knowledge  of  which  is  of  importance  in  practice.     Sup- 
pose a  body  of  earth  M,  Fig.  2,  supported  by  a' retaining  wall  AC, 
the  back  of  which  is  vertical.     Take 
Fig.  2.  as  a  first  case  that  the  earth  and  wall 

are  the  same  height,  and  the  earth's 
surface  in  no  way  extraneously  loaded. 
Suppose  that  a  wedge-shaped  piece 
ADE  separates  from  the  general  mass, 
and  thus  rests  on  the  retaining  wall  on 
the  one  side,  and  on  the  earth  on  the 
other ;  put  the  height  AD  of  the  earth 

and  wall  =  A,  the  density  of  the  earth  M=  y,  and  the  angle  AED 
which  the  surface  of  separation  AE  makes  with  the  horizontal  =$. 
Let  us  consider  a  length  of  the  mass  (at  right  angles  to  the  plane  of 
the  figure)  equal  unity,  then  the  weight  of  the  wedge  ADE : 

G-  =  AD^DE.  1  .  y  =  1  k  .  h  COtg.  *  .  y  =  1  h2  COtff.  *  .  y. 

The  vertical  back  A D  is  acted  upon  by  the  pressure  SP  =  P  at 
right  angles  to  it ;  and,  therefore,  it  may  be  assumed  that  an  equal 
opposite  horizontal  pressure  maintains  the  prism  ADE  on  the  in- 
clined plane.  We  know  also  (Vol.  I.  §  159)  that  a  force  will  be  taken 
up  by  a  bo4y-  if  its  direction  does  not  deviate  from  the  normal  to  the 
plane  of  contact  by  more  than  the  angle  of  repose,  and  we  may, 
therefore,  assume  that  the  second  component  force  R  of  G  is  taken 
up  by  the  mass  below  JlE,  even  supposing  its  direction  to  deviate 
from  the  normal  £7V  by  angle  RSN=  P.  As  NSG  =  AED  =  $,, 
we  have  RSG  =  $  —  P,  and,  therefore,  the  horizontal  pressure  on 
the  retaining  wall  P=  G  tang.  (<j>  —  P),  (compare  Vol.  I.  §  162),  or 
P=  I  h2  y  cotg.  $  tang.  (<?>  —  p). 

This  force  depends  upon  an  unknown  angle  $,  or  upon  the  dimen- 
sions of  the  prism  of  pressure,  and  is  thus  different  for  different 
values  of  <?>,  and  a  maximum  for  a  certain  value.  If,  now,  ADE 
be  the  prism  of  greatest  pressure,  and  ADO  a  prism  exerting  a 
less  pressure,  we  have  in  AEO  a  prism  which  requires  no  force  to 
maintain  it  on  its  basis,  but  which  would  rather  require  some  force 


PRESSURE  OF  EARTH.  15 

to  pull  it  downwards.  And  so  for  other  wedges  AOff,  &c.,  into 
which  we  might  divide  AEF,  because  these  rest  on  still  less  inclina- 
tions; we  may  therefore  assume,  that  by  an  opposite  force  equal  to 
the  maximum  pressure  P,  not  only  the  prism  ADE,  but  also  the 
prism  below  AE  and  AEF,  is  perfectly  sustained,  and  that  there- 
fore this  maximum  pressure  is  that  which  the  retaining  wall  is  sub- 
jected to  from  the  whole  mass. 

§  3.  Prism  of  greatest  Pressure. — We  must  now  determine  the 
prism  of  maximum  pressure.  We  have  manifestly  only  to  determine 
that  value  of  $  for  which  cotang.  $  tang,  (t — P)  is  a  maximum. 

Now  cotang.  *  tang.  (»— P)  =  g?'n'  (2  *  —  P)  —  ««*•  JP   and  as  this 

sin.  (2  <j> —  p)  +  sin.  p 

fraction  is  greater,  the  greater  sin.  (2  <j> — p)  is,  we  shall  have  cotang. 
<j>  tang.  (<?> — p)  a  maximum  when  sin.  (2  <j> — p)  is  a  maximum,  that 

is  =1,  or  2  t— p  =  90°,  i.  e.  $  =  45°  +  |.  Hence  we  name  the 
pressure  of  the  earth  against  the  retaining  wall: 

P=  i  A2  y  cotang.  /45°  +  p-\  tang.  /45°  —  |V 

or  since  cotang.  /45°  +  |)  =  tang.  ^45°  —  |V 

P  =  ±li2t[tang.  /45°  —  £\T. 
L          V  2/J 

The  complement  of  $  =  45°  +  i,  is  DAE=  45°  _^,  =  90°~p 


2  22 

=  one  half  of  DAF,  the  complement  to  90°  of  the  angle  of  friction  p. 
Therefore  the  surface  AE  of  the  prism  of  pressure  bisects  the  angle 
DAF  which  the  natural  slope  AF  makes  with  the  vertical  AD.  We 
can  now  very  easily  compare  the  pressure  of  a  disintegrated  or  semi- 
fluid mass  with  that  of  water.  In  the  latter  the  pressure  is  ^  h2  yt 
(Vol.  I.  §  276),  when  h  =  height,  1  =  breadth  of  the  pressed  surface. 
In  the  case  of  earth,  on  the  other  hand,  we  have  the  pressure 


where  7l  =  the  density  of  water,  and  «  the  specific  gravity  of  the 
semi-fluid.  Hence  the  pressure  of  earth  is  always 

t  tang.  (  45°  —  -  j  \  times  as  great  as  the  pressure  of  water,  or  the 
pressure  of  a  semi-fluid  may  be  set  as  equal  to  the  pressure  of  per- 
fect fluid  of  specific  gravity  £  \tang.  (  45°  —  -  )  I  . 

Thus  we  see  that  the  pressure  of  earth  increases  gradually  from 
the  surface  downwards,  or  is  proportional  to  the  pressure  -height. 

It  follows,  likewise,  that  the  centre  of  pressure  of  earth-works, 
&c.  &c.,  coincides  with  the  centre  of  pressure  of  water,  and  that, 
therefore,  in  the  case  in  question,  where  the  surface  is  a  rectangle, 
it  is  at  one-third  of  the  height  h  from  the  base  (Vol.  I.  §  278). 


16  COHESION  OF  SEMI-FLUID  MASSES. 

Example.  If  the  specific  gravity  of  a  mass  of  corn  seeds,  heaped  6  feet  high,  be  0,776 
(Vol.  I.  §  291,  remark  1),  it  exerts  a  pressure  against  each  foot  in  length  of  a  vertical 

wall:  P  =  i.6'.  0,776  .  63  .  [  (tang.  45°  —  15°)J  =  18  .63  .  0,776  (tang.  30°)'  = 
880  X  0,5773,5'  =  293|  Ibs.  (English.) 

§  4.  Cohesion  of  Semi-fluids.  —  In  the  above  investigations  we 
have  omitted  to  consider  the  cohesion,  or  that  mutual  union  of  the 
parts  of  the  mass,  increasing  with  the  surface  of  contact.  As  this 
cohesion,  however,  in  the  case  of  the  less  disintegrated  masses,  as, 
for  instance,  in  well  compacted  earth,  is  not  unimportant,  we  shall 
now  introduce  it  into  the  formula.  Let  us  put  the  modulus  of  co- 
hesion, or  the  force  of  union  for  the  unit  of  surface  of  contact  =  *, 
we  have  for  the  case  shown  in  Fig.  2,  the  force  required  to  separate 
the  prism  ADE  on  the  surface 

JE,  =  l.JlE.x=-^L. 

sin.  $ 

The  vertical  component  -^  —  sin.  $  =  x  h  counteracts  gravity,  and 

the  horizontal  component  --  cos.  <j>  =  x  h  cotq.  t,  counteracts  the 
sin.  <?> 

pressure.  If,  therefore,  we  introduce  into  the  formula  P=  G  tang. 
(t  —  p),  instead  of  P,  P+  x  h  cotang.  «j>,  and  instead  of  G,  G  —  x  h, 
we  then  obtain  the  equation  : 

P=(G  —  x  h)  tang,  (t  —  P)  —  x  h  cotang.  $. 
If  again  we  substitute  G  =  J  h2  y  cotang.  $,  we  have  : 

P  =  (i  A2  7  cotang.  9  —  *  '0  tang.  ($  —  p)  —  x  h  cotang.  $. 
It  is,  however,  convenient  to  make  the  following  transformations 
in  this  formula. 

P=  h  [(J  h  y  +  *  cotang.  P)  cotg.  <}>  tang.  ($  —  P)  —  *  cotang. 
¥—«(!  +  cotg.  *  cotg.  P)  tang,  (t  —  P)], 


tang.  (»-P)=    *«>*'*-  tang.  f 
1  +  tang.  $  tang.  P 


tana.  $  —  tang,  p 
--  —  - 


.  cotang.  ^  .  cotang. 


1  +  cotang.  $  cotang.  p 
we  have  P=h  [(%  hy  +  x  cotang.  p)  cotg.  <j>  tang.  (<j>  —  p) 

—  x  (cotg.  $  +  cotg.  P  —  cotg.  $)],  hence 
P  =  h  [(I  h  y  +  x  cotg.  p)  cotg.  $  tang.  (<?>  —  P)  —  x  cotang.  p]. 
•'.This  force  becomes  a  maximum  when  the  product  cotang.  $  tang. 
(»  —  P)  is  a  maximum,  and  as  we  have  seen  this  latter  is  so,  when 

<J>  =  45°  -f  ^;  therefore,  the  entire  horizontal  pressure  of  the  earth 

against  the  wall  : 

P  =  h  f(  £  h  y  +  x  cotg.  P)  Vtang.  (  45°  _  ^)T—  *  cotang.  Pl 


—  xh  cotang.  P  fl  —  Vtang.  (45°  —  i\T 


SURCHARGED  MASSES  OF  EARTH.  17 

2 

or  as  cotang.  p  =  --  —  ,  and 


tang.  (45°  +  0-ton0.  (45°  -0 


=  [tang.  (45°  +  0  -  tang.  (45°  _  0]  tarn,.  (45°  _  0, 
P  =  \  h2  y  Ytang.  (45°  —  0  J—  2  A  *  to/.  (45°  —  0 
-  A  tang.  (45°  -  0  [^  tang.   (45°  -  0  -  2  .  ]. 
This  force  is  0,  for  J  Ax  y  fcw0.  ^45°  —  0  =  2*,  that  is, 


4 

for  h    =  — 


^45°  —  0 


For  this  height,  therefore,  a  coherent  mass  may  be  cut  vertically, 
and  should  continue  so  to  stand.  -'.Inversely,  from  the  height  hl  of 
the  vertical  face  of  any  soil,  we  may  deduce  the  modulus  of-  cohesion, 
for 


h,  y  tang.  ^45°  —  0. 


Therefore,  the  cohesion  of  a  mass  is  greater  or  less,  according  to 
the  height  hl  for  which  it  maintains  a  vertical  face. 

If  we  introduce  1il  into  the  expression  for  P,  we  obtain  : 


For  sand,  seeds,  shot,  and  for  newly  turned  soils,  h1  is  very 
small ;  for  compressed  compact  soils,  it  is  sometimes  considerable  ; 
for  disintegrated,  moist  earth,  Martony  found  ^=0,9  feet,  whilst, 
for  the  same  material  soaked  with  water,  h1  =  0.  According  to  cir- 
cumstances, a  vertical  face  of  from  3  to  12  feet  maintains  itself  in 
different  soils.  > 

In  most  cases  of  practical  application,  it  is  advisable  to  omit  the 
effects  of  cohesion. 

§  5.  Surcharged  Masses  of  Earth. — If  the 
earth-work  JW,  Fig.  3,  be  loaded  on  the  sur-  Fte-  3- 

face,  with  buildings  or  otherwise,  as  DEH, 
the  retaining  wall  undergoes  an  increased 
pressure.  To  determine  this  increased  pres- 
sure, let  us  put  the  pressure  on  each  square 
foot  of  the  horizontal  surface  =  q,  then  the 
pressure  on  the  surface  for  ADE=q  .  DE 
=  qh  cotang.  <j>,  and,  therefore,  the  horizontal 
pressure,  without  reference  to  cohesion  : 

P  =  (G  4-  q  h  cotang.  t)  tang.  (<j>  —  p) 


18  RETAINING  WALLS. 

—  P)>  or  as  4,  =  45° 


To  find  the  point  of  application  of  this  force,  we  must  decompose 
it  into  its  two  parts  J  h2  y  f~tar#.  ^45°  —  ^  J  and 

9  A  l~fcm£.  /45°  —  j^T.     The  first  part  has  its  point  of  applica- 

tion at  $  of  the  height  h  above  the  base  A,  and,  therefore,  its  stati- 

cal moment  referred  to  this  point  : 

_*.!*.„  [W  (45=  _  I)]!  *Z  [W  (45o  _  I)  J  ; 

By  the  second  part,  however,  equal  portions  of  the  vertical  wall  are 
equally  pressed,  and  therefore  the  resultant  pressure  of  this  part 
passes  through  the  centre  of  gravity  of  the  wall,  and  acts  at  half 

the  height  -  from  the  base.  Hence  the  statical  moment  of  the  se- 
cond force 


The  moment  of  the  entire  pressure  is  thus  : 
(£  ft*  y  +  i  q  Ji^  rtang.  /45°  —  -  J  I  ,  and,  therefore,  the  leverage 

of  the  force,  or  the  distance  JW  =  a  of  its  point  of-  application  0 
from  the  base: 


8Ay-H 

Remark.— If  the  earth  be  carried  above  the  cope  of  the  wall,  and  form  from  it  a  natural 
slope,  the  formula  of  §  3  is  still  applicable,  if  h  be  put  equal  to  the  height  of  the  earth. 
and  not  that  of  the  wall. 

§  6.  Retaining  Walls. — The  pressure  of  earth  has  often,  in  engi- 
neering, to  be  withheld  by  retaining  walls  ( Fr. 
revetements),  or  by  walling -timbers,  and  sJteet- 
piling.  Retaining  walls  of  masonry  are  most 
usual,  and  we  shall,  therefore,  here  treat  of  these 
in  greater  detail. 

A  wall  AC,  Fig.  4,  may  be  either  pushed  for- 
ward, or  turned  over  by  a  force  KP=  P.  If  we 
suppose  this  wall  composed  of  horizontal  courses 
of  stone  bedded  on  each  other, 'we  may  assume 
that,  should  the  wall  give  way,  a  horizontal  crack 


SLIPPING  OF  WALLS. 


19 


will  form,  upon  which  the  upper  part  CU  slides  forward  or  turns 
about.  For  security  we  shall  neglect  the  effects  of  mortar,  and 
take  only  the  friction  between  the  beds'jjnto  consideration.  From 
the  force  P,  and  the  weight  G  of  the  part  CU  of  the  wall,  there 
results  a  force  KR  =  R  upon  the  magnitude  and  direction  of  which 
the  possibility  of  an  overturn  or  sliding  forward  of  this  part  of  the 
wall  depends.  If  the  angle  RKG,  by  which  this  resultant  deviates 
from  the  normal  to  the  plane  of  separation  UV,  be  less  than  the 
angle  of  friction  p,  the  wall  cannot  slide  forward  (Vol.  I.  §  159);  and 
if  the  direction  of  the  resultant  pass  within  the  joint  or  plane  of 
separation,  then  rotation  about  the  axis  V  is  not  possible  (Vol.  I. 

130). 

In  most  cases  of  application  it  will  be  found  that  rotation  more 
readily  takes  place  than  sliding,  and  therefore,  in  building  retaining 
walls,  provision  against  the  former  has  to  be  made.  Rotation,  or 
heeling  is  the  more  apt  to  occur,  in  as  much  as  it  not  unfrequently 
takes  place,  not  about  the  axis  F,  but  about  a  point  Vl  nearer  the 
resultant  R',  because  the  pressure  concentrated  in-  F,  compresses  or 
breaks  the  stone  near  the  point  F. 

If  the  points  of  intersection  H7,  for  a  series  of  resultants  R  pass- 
ing through  the  joints,  be  found,  and  a  line  drawn  through  these, 
we  have  what  is  termed  the  line  of -resistance,  and  it  is  easy  to  per- 
ceive that  an  overturn  of  the  wall  cannot  take  place,  so  long  as  this 
line  does  not  pass  beyond  the  joints  of  the  wall. 

If  the  force  P,  which  the  wall  has  to  withstand,  deviates  in  direc- 
tion from  the  vertical;  more  than  the  angle  p,  there  can  be  no  ques- 
tion of  its  sliding,  because  the  resultant  of  P  and  G  always  makes  a 
smaller  angle  with  the  vertical  than  P  alone. 

§  7.  Slipping  of  Walls. — If  we  substitute  for  P  the  pressure  of 
earth  found  above,x  we  can  determine 
the  thickness,  having  which,  a  wall  Fis-  5- 

will  be  sufficient  to  withstand  this 
pressure.  Let  us  consider,  in  the 
first  place,  the  case  of  slipping  for  a 
wall  JIC,  Fig.  5.  Suppose  that  the 
earth-work  pushes  forward  the  part 
UC,  on  the  joint  UV.  If  we  put  the 
thickness  at  top  of  wall  CD  =  b,  the 
relative  batter  =  w,  and  the  height 
DU  =  x,  we  have  the  thickness: 
UF  =  b  +  n  x,  and  the  contents  of  UC  for  1  foot  in  length 

//     l'~ 

=  b  x  -\ — __,  and,  therefore,  the  weight, 

G=  (fr  -\ — -  j  x  y^  yx  being  the  density  of  the  masonry.  For  the 
pressure  of  the  earth  on  DU,  we  have  generally: 

P  =  (i  x2  y  +  qx]  \  tang.  (45°  —  ^  )      ; 
\  -/J 


20 


ABUTTING  RESISTANCE  OF  EARTH. 


and  hence,  for  the  angle  RKG  =  <?>  made  by  the  resultant  R  with  the 
vertical : 

P          i  x2  y  +  q  x 
tang.  ?  =  —  *- 


—  -)     ;  or  as  t  must 


be  less  than  p,  therefore,  tang.  ^  <  /, 

_i£^_±_l_   .  [~tona.  /45°  —  ^\~f  <  /,  from  which  we  have  the 


thickness  of  wall  : 


For  x  =  0  we  have  the  thickness  at  the  top  : 

6  >  JL  Vtang.  (45°  —  iYT,  therefore,  for  $  =  0,  we  have  5=0; 

for  z  =  A,  the  whole  height  of  wall,  the  thickness  is  : 


0,  and 


To  apply  this  formula  to  a  dyke  or  dam,  we  must  put 

?=0;  then  we  get  b  >  (JL  __  n}-,  (Vol.  I.  §  280). 
Vli          /  2 

The  formulas  give  for  q  =  0,  (that  is,  when  the  surface  of  the  fluid 
or  semi-fluid  reaches  to  the  top  of  the  wall),  the  breadth  at  top  =  0  ; 
but  experience  has  proved  that  the  thickness  here  should  rarely  be 
less  than  2  feet,  and  in  positions  liable  to  wear  and  tear,  always 
above  this  dimension. 

Remark.  The  co-efficient  of  friction  for  stones  and  bricks  in  contact  with  each  other 
(Vol.  I.  §  161),  is  from  0.67  to  0,75.  And  when  a  bed  of  fresh-mortar  is  interposed,  only 
0,60  to  0,70.  Mortar  once  set,  acts  by  cohesion  or  adhesion,  and,  according  to  Boistanl, 
the  cohesion  of  mortar  is  from  800  to  1500  Ibs.  per  square  foot.  According  to  Morin. 
this  amounts  to  from  2000  to  5000  Ibs. 

§  8.  Abutting  Resistance  of  Earth.  —  We  must  distinguish  between 
the  active  and  passive  pressure  of  the  earth.  In  the  cases  hitherto 
considered,  the  pressure  is  active,  pressing  against  a  passive-resist- 
ance. The  pressure  of  earth,  however,  becomes  passive  when  it 
opposes  an  active  force  as  resistance,  as  when  it  resists  the  thrust  of 
an  arch,  &c.  &c.  Poncelet  has  termed  this  effect  of  earth-works 
butee  des  terres  (German,  Hebekraft  der  Erde),  and  Moseley  has 
termed  it  resistance  of  earth.  The 
resistance  which  a  body  opposes  to 
being  pushed  up  an  inclined  plane, 
is  greater  than  the  force  necessary  to 
prevent  the  sliding  of  the  body  down 
the  inclined  plane,  and  just  so,  in  the 
case  of  disintegrated  masses,  the  re- 
sistance which  they  oppose  to  a  ver- 
tical surface,  moved  horizontally,  is 


DEPTH  OF  FOUNDATIONS.  21 

greater  than  the  force  with  which  they  press  against  a  vertical  plane 
at  rest.  .'.Whilst  we  have  above  (Vol.  I.  §  162)  put  the  latter  force, 
P  =  G  tang.  ($  —  p),  the  resistance  of  the  iartter-  must  be  set  P  =  G 
tang.  (<?>  +  p),  or,  as  G  is  the  weight  J  h2  y  cotang.  $>)of  the  prism  of 
pressure  ADF,  Fig.  6,  P==^  h2  y  cotang.  $  tang.  (<j>+p).  .'.This  re- 
sistance P  depends  on  the  angle  AFD  =  $  at  which  the  assumed 
plane  of  separation)  intersects  the  horizontal,  and  is  a  minimum  for 
a  certain  value  of  <?>.  But  in  order  to  find  this  value,  let  us  put : 

cotang.  *  tang,  fa  +  P)  =  sm'  ^  *  "j"  p| +  sm'  p, 
sin.  (2  <?>  +  p)  —  sin.  P 

and  we  see  at  once  that  this  is  a  minimum,  when  sin.  (2  $  +  p)  is  a 
maximum,  that  is  when 

2  $  +  p  =  90°,  therefore,  *  =  45°  _  |. 

If  we  now  introduce  this  value  into  the  formula  for  P,  we  obtain 
the  least  resistance  of  the  earth-work. 

P  =  J  W  y  cotang.  (45°  _  0  tang.  (45°  +  0 

=  i 

This  is,  generally,  the  resistance  with  which  earth  or  any  other 
disintegrated  mass  withstands  a  moving  force;  for  as  soon  as  this 
force  is  equal  to  that  resistance,  a  yielding  of  the  mass  takes  place. 

§  9.  Depth  of  Foundations. — An  important  application  of  the 
passive-resistance  of  earth,  arises  in  the  founding  of  retaining  and 
other  walls.  If  the  ground  on  which  the  retaining  wall  is  to  stand 
be  clayey,  or  wet,  the  co-efficient  of  friction  between  the  wall  and  the 
ground  may  fall  as  low  as  0,3,  and  then 
a  slipping  of  the  wall  may  very  easily  Fig.  7. 

occur.  It  is,  therefore,  necessary  in 
such  cases  to  dig  the  foundation  to  such 
a  depth  that  the  passive -resistance  on 
the  outside,  combined  with  the  friction 
on  the  bottom,  may  counterbalance  the 
active -pressure  on  the  inside.  If  G  be 
the  weight  of  a  supporting  wall  AC, 
Fig.  7,  therefore fG  its  friction  on  the 
bottom,  AB;  if  h  be  the  height  of  the 
earth  at  the  back,  and  ht  the  height  in 

front;  if  further,  p  and  y  be  the  angle  of  friction,  and  the  density 
for  the  one,  and  Pl  and  7l  those  for  the  other  earthy  mass,  we  have : 

'45°  +  Pj-)T=  \K  V 
and  therefore  the  depth  BK  of  the  foundation  for  such  a  wall: 


22 


HEELING  OF  RETAINING  WALLS. 


For  security,  a  co-efficient  of  stability  1,4  has  been  introduced  (by 
French  engineers  for  the  revetement  walls  of  fortifications),  and 
therefore  the  depth:  _ 


is  given  to  such  walls. 

Example.  To  what  depth  must  a  parallel  wall  8  feet  thick,  and  13  feet  clear  height, 
have  its  foundation  sunk,  that  it  may  withstand  the  pressure  of  water  standing  level 
with  the  top  of  the  wall?  In  this  case  j  ^  0,  y  =  62,25  Ibs.  (for  which  we  take  03) 
h  =  13  feet;  also  /=  0,3,  fl  =  30°,y=  1,6  X  63  =  100,8  Ibs.,  and  G,  (the  density  of  the 
masonry)  being  2  X  *>3  =  126  Ibs.,  must  be  =  8  X  13  X  126  =  13104  Ibs.,  therefore, 


A,  =  1,4  fang.  (45°  — : 


13"  X  63  —  2x0,3  X  13104 


4,25  feet  very  nearly. 


§  10.    Heeling  of  Retaining  Walls. — In  order  to  appreciate   a 
retaining  wall   in   reference   to   stability,   it   is   necessary  to    de- 
termine  its    line   of  resistance^.     For  simplicity, 
we  shall  first  take  a  parallel  wall  AC,  Fig.  8.     If 
we  had  only  a  horizontal  force  KP  =  P  to  deal 
with,  the  point  of  application  of  which  is  at  a  dis- 
tance D0=a  from  the  cope  of  the  wall,  the  line 
of  resistance  would  be  a  hyperbole,  as  the  follow- 
ing simple  view  of  the  subject  shows.    Of  the  force 
P  (whose  point  of  application^we  assume  in  the 
line  passing  through  the  centre  of  gravity >of  the 
wall)  and  the  weight  G  of  UVCD,  the  resultant  is 
R  which  intersects  UV  in  W,  a  point  in  the  line  of 
resistance  sought.     If  we  now  put  the  thickness  of  the  wall  AB  = 
CD  =  b,  its  density  =  yx,  the  abscissa  KN=  x,  and  the  ordinate 
J\W=  y,  we  have  G  =  (a  -f  x}  b  y1?  and  from  similarity  of  tri- 
angles : 

KWJY  and  KRG:  —  =  — ,  that  is  y-  = - , 

KN       KG  x      (a  +  x}  b  Tl 

and  hence  the  equation  of  the  line -of-  resistance  y  = —7 

(a  +  x)  b  yx 

0,^=0,   and  for  x  =  ex, 
The  curved  line -of  resist- 


From   this   we   see   that  when 

y  —  - — ,  and  for  x  =  —  a,  y  =  —  oo. 

6yj 

ance,   therefore,   passes   through  K,    and 
'-•  9-  has  not  only  the  horizontal  CD,  but  like- 

wise a  vertical  EF  for  asymptote,  distant 

p 
ST  = from  the  centre  of  gravity  S  of 

the  wall.  1 

It  is  otherwise,  of  course,  for  a  wall  to 
withstand  pressure  of  earth  or  water  as 
AC,  Fig.  9,  for  here  a  is  variable,  because 
P  is  applied  at  a  point  U  at  J  of  the  height 


HEELIXG  OF  'RETAINING  WALLS.  23 

DfJfrom  the  base.    If  we  draw  the  end  of  the  vertical-line  through 
S  as  origin  of  the  co-ordinates,  that  is,  if  we  put  HN=  x,  we  hav*e: 


y  =  rr|—  [tang.  /45°  —  *L\~\   •  x*-     Tnis  equation  corresponds  to 

O  0  yl  L  */  J 

the  common  parabola  with  absciss  ?/  and  ordinate  a;. 

If,  however,  we  suppose  the  earth-work  carried  a  height  ht  above 
the  cope  of  the  wall,  we  must  adopt  the  proportion  : 

^        =  il-  [tang.  (45°  —  ^)"f(x  +  Ax)2,  whence  we  have  the 

S  \x  ~^~  "i)       0  #  7i  L  \  -^/J 

equation  y  =  _L_  fte^.  ^45°  -  i\TJfe±*£, 
66y,L          V  2/J         a; 

§  11.  The  stability  of  a  retaining  wall  requires  not  only  that  the 
line-of-  resistance  be  within  the  wall,  but  also  that  it  shall  not  come 
too  near  the  outside  of  it.  The  famous  Marshal  Vauban  gives  the 
practical  rule  :  that  the  line-of  resistance  should  intersect  the  basis 
of  the  wall  in  a  point  whose  distance  from  the  vertical  passing 
through  the  centre  of  gravity  of  the  wall  is  at  most  |  of  the  distance 
of  the  outer-  axis  of  the  Avail  from  this  line.  If,  as  Poncelet  does, 

FB  r~ 
we  call  the  reciprocal  of  this  number,  or  the  ratio  -  between  the 

distance  of  the  outer  axis  from  the  vertical  passing  through  the 
centre  of  gravity,  and  the  distance  of  the  point  of  intersection  L 
of  the  line  of  -resistance  from  this  gravity  line,  .  the  co-efficient  of 
stability,  and  represent  it  generally  by  8,  we  have  for  the  stability 
of  a  parallel  wall,  withstanding  the  pressure  of  earth,  (by  introducing 
into  the  last  formula  instead  of  x,  the  height  h  of  the  wall,  and 

instead  offti-V 


and,  therefore,  the  requisite  thickness  of  the  wall  : 


If  for  8  we  substitute  f  =  2,25,  and  for  —  ,   f   a  mean  value,  we 

Ti 

get:  _ 

b  =  0,707  (h  +  h,}  J^A  •  tang.  (45°  _  0. 

If  we  take  P  =  30°,  we  obtain  6  =  0,4  (h  -f  A,)     I  A±A  . 
Poncelet  gives  : 

b  =  0,865  (h  +  hj  tang.  /45°  _  p-\     I-I,  or  approximately  : 
b  =  0.285  (h  +  AJ,  for  cases  in  which  h^  varies  from  0  to  2  A. 


24  PONCELET'S  TABLES. 

Example.  What  must  be  the  thickness  of  a  parallel  wall  of  28  feet  in  height  to  retain 
broken  stones,  mine  rubbish  for  a  height  of  35  feet?  Assuming  that  the  density  of  the 
\vall  =  2,4  X  63  =  151,2  Ibs.  The  density  of  the  rubbish  1,3  X  63  =  81,3  Ibs.,  and 
^  =  50°.  According  to  Poncelet's  formula: 


0,865  x  35  tang.  (45°  —  25°)      I—  =  30,3     I—  .  tang.  20°: 


=  8,11  feet. 


§  12.  Poncelet's  Tables.  —  To  facilitate  applications  of  the  for- 
mula, Poncelet  has  calculated  the  following  table,  which  contains 

values  of  -  corresponding  to  given  values  of  -i,  2-,  and  p.      There 

h  h    yj 

are  two  cases  distinguished  in  the  table,  namely,  the  case  when  the 
earth-work  is  heaped,  as  is  shown  in  Fig.  7,  the  coping  being  covered, 
and  the  case  shown  in  Fig.  10,  where  a  berme  of  the  breadth  0,2  7i, 
from  the  outer  edge  of  the  cope  of  the  wall,  is  left  before  the  natural 
slope  of  the  embankment  begins  :  so  that,  in  short,  a  promenade  is 
left  of  the  width  C£=0,2  A. 

The  headings  of  the  table  explain  themselves. 


PONCELET  S  TABLES. 


25 


1  * 


COt—  O1OO 
t— OOCNt-COCl  Oi— i  CO  O 
to  CD  t-  t-  t- 


QOtOOS(MO5CNCCOTt<CO<MtOr- IT— I 


t—  O  (M  Tt<  iO  CD  t- 


\  N 


Ttl<MTHCOa5OO(MOO^ 

tOCDt— GOCirHCOCOlO 


i-   r-   r-   r- 


i— ICDODt— COrt< 


VOL.  II.— 3 


26  RETAINING  WALLS  WITH  BATTER. 

In  this  table  the  limiting  values  have  been  principally  held  in 

view.     Thus  —  =  1  corresponds  pretty  nearly  to  one  limit  of  the 
v 

ratio  of  the  densities  of  masonry  and  earth,  while  —  =   J   corre- 

sponds to  the  other.  Again,  /  or  tang.  P  =  0,6,  is  the  value  for  the 
least  coherent  earths,  and  /=  1,4  the  value  for  stiff  compact  earth- 
work. In  many  practical  cases,  the  required  proportion  has  to  be 
deduced  by  interpolation. 

Remark.  The  formula  b=  .  865  (A+  A,)  tang.  ^45°  —  -L\  /*_  gives^  results  corre- 
sponding with  those  in  the  table  to  within  T'5. 

The  values  in  the  table  refer  to  parallel  walls,  built  with  mortar. 
If  the  external  batter  of  the  wall  does  not  exceed  0,2|  that  is  2,4 
inches  per  foot,  the  breadth  b  found  above,  will  be  that  of  the  wall 
at  ^  of  its  height  from  the  base,  and  through  this  point  the  line  of 
batter  is  to  be  drawn. 

Remark.  The  dimensions  resulting  from  Poncelet's  rules  or  tables  are  applied  in  France 
for  walls  of  fortifications,  but  give  dimensions  nearly  one-fourth  greater  than  the  average 
practice  of  civil  engineers  in  Britain  for  the  same  relative  circumstances.  —  TH. 

§  13.  Retaining  Walls  with  Batter.—  If 
lg'  *  '  _  the  wall  has  a  batter,  or  if  its  profile  be  a 
trapezium  AC,  Fig.  10,  the  thickness  neces- 
sary to  insure  resistance  to  rotation  can 
only  be  determined  by  aid  of  a  complicated 
expression.  If  we  assume  the  face  AB  as 
the  plane  of  separation,  and  put  KF=  OA 
=  x,  and  FL  =  y,  we  then  have  again  : 

i«a 

G 


1),  and 
G 


y  [W  (45°  -  |)J, 


P  =  H*  + 

and  if  I  the  thickness  at  top,  and  n  the  relative  batter,  therefore, 
n  h  the  absolute  batter,  G  =  (bh  +  J  nh2}  7l,  and,  therefore, 


7l  '  6  h  (b  +  1  nh) 

The  distance  BF  of  the  outer  edge  of  the  wall  from  the  vertical 
passing  through  the  centre  of  gravity,  is  : 

=  b  +  nh       3b  +  nh    nh       362  +  6n  hb  +  2  n*h2 

2 
and  hence  we  may  put  : 

3  b*  +  6  n  hb  +  2 


n2  h2  =1?'  .  (A+_M3  T^.  /45°  _  t\T 


ARCHES.  27 

and  hence  the  thickness  of  wall  at  top : 


Remark  1.  If  the  back  of  the  wall  have  a  batter  likewise,  we  have  a  different  prism 
of  greatest  pressure  to  deal  with,  because  the  force  applied  to  the  wall  is  no  longer  hori- 
zontal. The  investigation  becomes  complicate,  and  we  forbear  to  enter  upon  it,  but  shall 
refer  to  works  treating  of  the  subject. 

Remark  2.  Coulomb  was  the  first  to  propound  a  good  theory  of  the  pressure  of  earth. 
See  "  Theorie  des  machines  simples."  Prony,  in  his  "Le9ons  sur  la  poussee  des  terres, 
(1802,)"  extended  Coulomb's  theory.  Navier  pursues  the  same  notions,  with  much  ele- 
gance and  precision,  in  his  "Lemons  sur  1'application  de  la  mecanique,  tome  i."  May- 
niel,  in  1808,  published  a  special  treatise  on  the  pressure  of  earth,  in  which  the  observa- 
tions and  theories  of  his  predecessors  are  reviewed,  "  Traite  experimental,  &c.,  de  la 
poussee  des  terres."  C.  Martony  de  Koszegh  made  experiments  on  a  large  scale  for  the 
Austrian  government,  which  were  published  in  1828,  under  the  following  title:  "  Ver- 
suche  uber  den  Seitendruck  der  Erde,  ausgefuhrt  auf  hochsten  Befehl,  &c.,  und  verbunden 
mit  den  theoretischen  Abhandlungen  von  Coulomb  und  Frangais,  Wien,  1828."  The 
most  complete  work  on  the  pressure  of  earth  is  that  of  Poncelet  in  the  "  Memorial  de 
1'officier  du  genie,  1838,"  and  which  has  been  translated  into  German  by  Lahmeyer, 
Braunschweig,  1844.  In  Moseley's  "Engineering  and  Architecture,"  this  subject  is 
handled  with  great  elegance  and  success.  Hagen  has  a  chapter  on  this  subject  in  the 
second  part  of  his  admirable  "  Wasserbaukunst,"  in  which  he  takes  a  peculiar  view 
of  it. 


THEORY    OF    ARCHES. 


§  14.  Arches. — An  arch  (Fr.  votite,  Ger.  G-ewolbe],  is  a  system  of 
bodies  resting  upon  each  other,  and  supported  by  two  fixed  points, 
in  such  manner  that  they  are  in  equilibrium  not  only  among  them- 
selves, but  with  certain  external  forces.  The  material  of  these  bodies 
is  usually  stone,  and  hence  are  termed  arch-stones  (Fr.  voussoirs, 
Gr.  G-ewolbesteine}.  The  planes  of  contact  of  the  stones  are  the 
beds  or  joints.  The  fixed  points  upon  which  the  arch  rests  are 
termed  abutments  (Fr.  Pieds-droits,  Ger.  Widerlager),  and  in  cer- 
tain cases  piers  (Fr.  culees,  piliers,  Ger.  Pfeiler).  Of  the  arch- 
stones,  the  highest  is  termed  the  key -stone  (Fr.  clef,  Ger.  Schluss- 
stein\  and  those  which  rest  on  the  abutments  or  piers,  are  termed 
imposts  or  springers*  (Fr.  coussinets,  Ger.  Kdmpfer.}  An  arch  is 
included  between  two  more  or  less  curved  surfaces,  the  intrados  and 
extrados,  which  are  sometimes  termed  the  soffit,  and  the  back  of  the 
arch. 

As  regards  the  intrados  and  extrados,  arches  are  very  various. 
Cylindrical  surfaces  are  most  usual,  but  conical  surfaces  occur,  and 
we  have  domes,  and  variously  proportioned  groinings.  We  shall 
treat  of  cylindrical  arches  only,  and  limit  ourselves  still  further,  to 
the  consideration  of  those  having  a  horizontal  axis.  Such  arches 
are  bounded  by  two  vertical  parallel  planes,  the  faces  of  the  arch, 


28 


LINE  OF  PRESSUBE  AND  RESISTANCE. 


(Fr.  parements,  Ger.  Stirnflachen.}  According  as  the  faces  are 
perpendicular  or  inclined  to  the  geometrical  axis  of  the  arch,  the 
arch  is  direct,  or  oblique,  or  skewed  (Fr.  droites  or  biaises);  groined 
arches  or  vaults  (Fr.  votites  d 'arete,  Ger.  Kreuz,  or  Klostergewolbe], 
are  merely  intersecting  cylindrical  arches.  Domes  or  cupolas  (Fr. 
voutes  en  dome,  Ger.  Kuppel  or  Kesselgewolbe),  are  arches  generated 
by  the  revolution  of  a  curve  about  a  vertical  axis. 

As  regards  the  curvature  of  arches,  it  is  very  various.  The  sec- 
tion is  sometimes  circular,  sometimes  elliptical,  catenarian,  or  formed 
of  several  circular  arcs,  and  plate  bands,  or  straight  arches  are  some- 
times built. 

Remark. — As  experience  has  abundantly  proved  that  arches  fail  or  give  way  by  a 
rotation  of  determinate  parts  round  the  edges  where  certain  joints  meet  the  extrados  or 
intrados,  and  not  by  sliding  dislocation,  we  need  here  only  consider  the  conditions  of 
equilibrium  in  reference  to  the  former  circumstance,  omitting  our  author's  investigation 
of  the  latter,  which  show,  as  is  usually  done  in  elementary  treatises  of  mechanics,  that 
for  the  case  of  equilibrium  without  friction,  the  weight  of  the  arrh-stones  nntst  be  to  each 
other  as  the  differences  of  the  cotangents  of  the  angles  of  inclination  of  the  joints  to  the  hori- 
zon.— TH. 

Remark. — The  dislocation  of  an  arch  by  slipping  of  voussoirs  might  occur  in  two 
ways:  according  as  the  joint  of  maximum  pressure  lies  above  or  below  the  joint  of 
minimum  pressure.  In  the  former  case,  Fig.  11,  the  hauches  of  the  arch  slide  out,  and 

Fig.  11. 


the  crown  slips  down.     In  the  other  case,  the  reverse  happens.  Fig.  12.     This  second 
case  scarcely  ever  occurs,  so  that  we  shall  not  farther  recur  to  it. 


Fig.  13. 


§  15.  Line  of  Pressure  and  Re- 
sistance.— An  arch  is  so  much  more 
likely  to  fall  in  by  rotation  round 
the  outer  or  inner  edge  of  a  joint, 
than  by  slipping,  that  the  former 
may  be  considered  as  the  usual  ac- 
cident. The  stability  of  an  arch 
in  reference  to  rotation  may  be 
considered  exactly  in  the  same 
manner  as  the  stability  of  a  pier 
or  Avail  (Vol.  II.  §  6).  From  the 
horizontal  force  Pl  applied  at  any 
point  0,  Fig.  13,  in  the  crown  of 
the  arch  and  the  weight  of  the  first 
arch-stone  acting  in  its  centre  of 
gravity  Sv  there  results  the  force  P2  acting  on  the  first  joint,  and 
the  intersection  01  of  the  direction  of  this  force  with  the  joints  El  -Fr 


LINE  OF  PRESSURE  AND  RESISTANCE. 


29 


Again,  from  the  pressure  P2,  and  the  weight  G2  of  the  second  arch- 
stone,  acting  in  its  centre  of  gravity  S2,  there  results  the  pressure  P3 
in  the  second  joint,  and  the  intersection  02  of  the  direction  of  this 
force  with  the  second  joint.  Proceeding  in  this  manner,  we  obtain 
the  remaining  normal  pressures,  and  the  intersections  03,  04,  &c., 
in  the  other  joints.  But  the  lines  0,  015  02,  03  .  .  .  ,  which  unite 
the  intersections  or  points  of  application  of  the  pressures  P1?  P2, 
P3 .  .  .  ,  is  the  line  of  resistance  (Fr.  ligne  de  Pression,  Ger.  Wider- 
standslinie),  (Vol.  II.  §  6).  So  long  as  at  least  one  line  of  resistance 
can  be  found  in  the  face  of  an  arch,  which  neither  passes  beyond 
the  intrados  nor  the  extrados  at  any  point,  so  long  dislocation  of  the 
arch  by  rotation  cannot  occur.  If,  on  the  jpther  hand,  the  line  of 
resistance  intersects  the  intrados,  the  arch  "will  fall  inwards,  and  if 
it  goes  beyond  the  extrados,  the  arch^will  rise  upwards,  and  so  fall 
to  pieces.  Fig.  14  represents  the  former  case,  and  Fig.  15  the  latter. 


Fig.  14. 


Fig.  15. 


Fig.  16. 


The  dislocation  becomes  inevitable,  however,  from  the  circumstance 
that  the  incompressibility  of  the  stones  opposes  resistance  to  the 
forces  RR,  acting  with  the  leverages  £0,  FO.  The  cohesion  of  the 
mortar  alone  resists  this  force;  but  as  a  very  slight  concussion  is 
sufficient  to  destroy  this  cohesion,  its  effects  should  not  be  relied 
upon  as  available. 

It  is  easy  to  perceive  that  arches  are  so  much  the  more  stable  (in 
reference  to  rotation)  the  greater  the  number  of  lines  of  resistance 
that  can  be  drawn  within  them; -the  less, 
therefore,  the  number  of  lines  of  resistance 
that  intersect  the  intrados  or  extrados.  The 
arch  of  greatest  stability,  Fig.  16,  is  neces- 
sarily that  in  which  a  line  of  resistance  may 
be  drawn,  which  passes  through  the  centre 
of  all  the  arch-stones,  or  bisects  their  depth. 
For  the  usual  construction  of  arches,  that 
is,  for  circular  arches,  a  rotation  or  rising 
upwards,  that  is,  an  intersection  of  all  lines 
of  resistance  with  the  extrados;  cannot  possibly  occur;  we  may,  there- 
fore, limit  ourselves  in  the  investigation  of  stability  to  the  rotation 
from  which  the  arch  falls  inwards.  That  we  may  be  certain  that  at 
least  one  line  of- resistance  passes  beyond  neither  intrados  nor  ex- 

3* 


30          EQUILIBRIUM  IN  REFERENCE  TO  ROTATION. 

trades,  we  may  start  to  draw  it  from  the  point  D  in  the  crown,  and 

try  whether  it  intersects  the  intrados. 

§  16.  Equilibrium  in  Reference  to  Rotation. — The  conditions  of 
stability  in  reference  to   rotation  may 
Fig.  17.  be  considered  in  another  point  of  view, 

and  one  more  adapted  for  calculation. 
We  may  eliminate  the  forces  Pv  P2, 
P3 . . .,  acting  in  the  crown  D,  Fig.  17, 
which  are  necessary  to  hinder  a  rotation 
of  the  arch-stones  round  the  inner  edges 
Ev  E2,  E3,  &c.,  and  then  investigate 
which  is  the  greatest  of  these  forces. 
If  we  designate  the  leverages  E^L^ 
E2L2,  E3L3, ...  of  the  force  P  referred 
to  the  points  Ev  E2,  E3,  &c.,  as  axis  of 
rotation  by  ar  a2,  a3,  &c.,  and  the  lever- 
ages E1H1,  E2H2,  E3H3,  &c.,  of  the 

weights    G,,   Gl  +  G2,  Gv  +  G2  +  G3,   &c.,  in  reference  to  these 

same  axis  by  bv  b2,  b3,  &c.,  we  have  for  the  force  P  acting  at  the 

crown : 

P,  =  *L  Gv  P2  =  12(G.  +  G2),  P3  =  -3  (G,+  G2+  G3),  &c. 

a,  a2  a3\ 

But  not  only  the  factors  bn ,  and  Gx  +  G2  +  . . .  +  Gn  of  the  nume- 
rator increase  from  the  crown  towards  abutments,  but  the  denomi- 
nator Jin  increases  also ;  hence  one  of  the  values  of  Pv  P2,  P3,  &c., 
is  a  maximum ;  and  it  is  necessary  for  equilibrium,  that  the  effective 
force  Pm,  acting  in  the  crown,  should  be  equal  to  it.  That  joint 
which  corresponds  to  the  maximum  -pressure,  or  the  pressure  on  the 
crown,  is  termed  the  joint  of  rupture  (Fr.  joint  de  rupture,  Ger. 
Bruchfuge],  because  dislocation  by  rotation  first  begins  round  its 
lower  edge,  if  the  force  Pm  at  the  crown  diminishes.  It  is  deter- 
mined by  the  angle  of  rupture,  which  its  plane  makes  with  the 
horizon  (or  with  the  vertical).  It  is  also  easy  to  perceive,  that  the 
angle  of  rupture  gives  that  point  in  the  arches,  in  which  the  line 
of  resistance,  starting  in  D  in  the  crown  of  the  arch,  touches  the 
intrados.  fc4Lj,  j^fc 

If  we  compare,  the  maximum- effort  required  to  hinder  rotation 
inwards  with  the  maximum- effort  required  to  resist  slipping,  we  find 
that  in  most  cases  the  force  required  to  resist  rotation  is  greater 
than  that  to  resist  slipping,  and,  therefore,  the  pressure  in  the  crown 
of  an  arch  is  equal  to  the  greatest  of  all  the  forces  Pv  P2,  P3,  $c., 
which  oppose  the  rotation  of  the  parts  of  the  arch  Gv  G1  +  G2, 
Gj  +  G2  +  G3,  £c.,  round  the  inner  edges.  If,  therefore,  we  have 
once  determined  this  pressure  at  the  crown  of  the  arch,  it  is  easy  to 
find  the  pressure  in  any  other  part  of  the  arch. 

Arches  falling  by  rotation  outwards  are  exceptional  cases.  To 
discriminate  by  calculation  as  to  the  possibility  of  such  an  accident 
occurring,  the  point  of  application  of  the  force  P  is  taken  at  the 


STABILITY  OF  ABUTMENTS. 


31 


lower  edge  ./?,  Fig.  18,  of  the  joint  of  the  key-stone,  because  the 

leverage,  in  reference  to  rotation  about 

Fv  F2,  F3,  &c.,  is  here  the  least.    If  now  Fig.  18. 

we  again  designate  the  leverages :  F^L^ 

F2L2,  F3LZ,  &c.,  by  av  av  «3,  &c.,  and  the 

leverages  F^v  F2H2,  F3H5,  &c.,  of  the 


weights 


G 


we  have  the  values  o 


,     , 
f  P  : 


G2+  G3, 


and  if  the  least  of  these  values  be  greater 

than  the  pressure  in  the  crown,  or  the 

greatest   of  the   forces  which  prevent  a 

falling  inwards,  the  arch  is  stable  ;  unless  this  be  the  case,  disloca- 

tion takes  place. 

Remark.  The  falling  to  pieces  of  an  arch  by  rotation  may  likewise  happen  in  two 
ways:  according  as  the  joint  of  rupture  of  the  maximum  value  is  above  or  below  the 
joint  of  rupture  of  the  minimum  value.  Fig.  19  represents  the  first,  and  Fig.  20  the 
second  case. 


Fig.  19. 


Fig.  20. 


Fig.  21. 


§  17.  Stability  of  Abutments. — If  we  have  satisfied  ourselves  by 
the  calculations  indicated  in  the  forego- 
ing paragraphs,  that  an  arch  is  stable, 
and  have  also  determined  the  pressure 
in  the  key-stone,  we  have  still  to  inves- 
tigate the  stability  of  the  abutment 
walls;  that  is,  to  determine  the  thick- 
ness of  abutment  wall  necessary  to  re- 
sist a  detrusion  or  an  overturn.  This 
investigation  is  the  more  important,  as 
it  is  not  unfrequently  in  consequence 
of  insufficient  resistance  of  these,  that 
arches,  in  themselves  stable,  fall  in. 

It  is  evident  that  a  retaining  wall 
FB,  Fig.  21,  is  stable  when  the  direc- 
tion of  the  resultant  force  Kl  ^  =  Rj^ 
of  the  weight  of  the  one  semi-arch  act- 
ing at  its  centre  of  gravity  S,  the  hori- 


32  STABILITY  OP  ABUTMENTS. 

zontal  thrust  P  acting  at  the  crown,  and  Sv  the  weight  of  the  re- 
taining wall,  passing  through  its  centre  of  gravity,  passes  through 
the  base  FO  of  the  retaining  or  abutment  wall,  and  deviates  from 
the  vertical  K^  by  an  angle  less  than  the  angle  of  repose  P. 


For  the  angle  <|>,  which  the  resultant  R^  of  the  forces  P  =  KP1  and 

G  +  Gj  =  K  G^  makes  with  the  vertical,  we  have  tang.  <j>  =  -  ; 

G  +  Gl 

but  tang,  p  =  the  co-efficient  of  friction  /,  and  hence  to  insure  sta- 

p 

bility  in  reference  to  sliding,  we  should  have  -  <  /. 

G+  Gj 

In  order,  further,  that  the  resultant  may  pass  through  the  outer 
edge  .Fof  the  abutment,  let  us  put  the  moment  of  P,  referred  to  this 
edge,  equal  to  the  sum  of  the  moments  of  the  weights  G  and  Gr 
If  a  be  the  rise  of  the  arch  BL,  and  h  the  height  of  the  abutment, 
then  the  moment  of  the  force  P  referred  to  the  edge  F  as  an  axis 
=  P  (a  4-  h);  if,  again,  b  be  the  horizontal  distance  BH  of  the  ver- 
tical passing  through  the  centre  of  gravity  of  the  semi-arch  J3C  from 
the  inner  edge  B  of  the  springing  point,  c  the  thickness  of  the  abut- 
ment wall,  and  e  the  distance  FN  of  the  vertical  gravity-line  of 
the  abutment  wall  from  the  edge  F,  we  have  the  moment  of  the 
weights  G  and  G1  =  G  (b  +  e)  +  G^,  and  thus  we  get  the  equa- 
tion: P  (a  +  h)  =  G  (b  +  c)  +  Gl&. 

In  order  to  insure  permanence,  experience  dictates,  according  to 
Audoy's  deductions,  the  employment  of  1,9  P  instead  of  P,  so  that 
the  equation  for  determining  the  thickness  of  the  abutment  becomes: 
1,9  P  (a  +  h)  =  G  (b  +  c)  +  G1&.  If  h,  be  the  mean  height  of  the 
abutment  or  pier,  and  y  the  density  of  its  masonry,  we  have  for  each 
foot  in  length  of  the  pier  the  weight  Cr1=hlc  y,  and  if  we  put  e  =  \  c, 
the  moment  Q-^e  =  1  Ax  c2  y,  and  hence: 

J  \  c2  y  +  G  c  =  1,9  P  (a  +  h)  —  a  5,  or, 
cZ  .   2ffc  _  l,9P(q  +  A)  —  G-b 

r*      7  _    7  9 

"'l  V  ^  MI  7 

and  hence  the  thickness  of  the  abutment  in  question  : 
G_         fl,9P(a+Ij~—  ab   ,    /  # 

^  r  +  W          ~T^7~  f  \  h 

In  order  to  secure  this  wall  against  sliding,  we  must  have  : 


i 

It  will  usually  be  found  that  the  former  value  of  c  is  greater  than 
the  latter;  and  that,  therefore,  the  thickness  of  the  abutment  must 
be  regulated  by  the  former  condition  of  stability. 

For  very  high  piers,  as  G  c,  1,9  P  a  and  G  b,  are  very  small  com- 
pared with  1,9  Ph  and  J  \  c2  y  (which  may  be  put  \  h  c2  y),  we  have: 
J  h  c2y=l,9  P  A,  i.  e.  J  c2  y  =  1,9  P,  and  hence  the  greatest  strength  : 


LOADED  ARCHES. 


33 


§  18.  Loaded  Arches. — We  have  hitherto  neglected  to  consider 
the  influence  of  the  backing  on  the  arch ;  which,  however,  it  is  essen- 
tial to  examine.  That  the  stability  of  an  arch,  such  as  a  bridge, 
may  not  be  altered  by  the  passage  of  heavy  weights  upon  it,  it  is 
necessary  that  the  arch  should  in  itself  possess  such  weight,  or  be 
permanently  loaded  with  backing,  that  any'weight  arising  from  traffic, 
such  as  heavy  wagons,  locomotives  and  the  like,  can  only  occasion  a 
slight  change  in  the  entire  load,  or  forces  in  action. 

The  backing  consists  usually  of  a  system  of  walling  (spandril 
walls),  supporting  the  road-way,  and  carried  up  either  to  form  a 
horizontal  line  EF,  Fig.  22,  or  an  inclined  line,  Fig.  23.  In  very 


Fig.  22. 


Fig.  23. 


many  cases,  the  spandril-walls  or  backing  of  arches  consist  of  the 
same  materials  as  the  arches;  and  if  it  be  uniformly  built,  we  may 
assume  a  common  density  for  the  whole,  and  thus  considerably  ab- 
breviate calculation.  If,  according  to  Vol.  I.  §  58,  we  take  the  spe- 
cific gravity  of  masonry  at  from  1,6  to  2,4,  we  have  for  the  density 
of  the  masonry  100  to  150  Ibs.  per  cubic  foot,  the  former  answering 
to  brick-work,  the  latter  to  ashlaring.  The  loading  of  arches  gene- 
rally increases  their  thrust,  and  also  their  stability.  That  the  vous- 
soirs  may  resist  crushing,  they  must  have  a  certain  depth  propor- 
tioned to  the  pressure  of  the  arch;  and  as  this  increases  from  the 
crown  towards  the  springing,  the  depth  of  the  voussoirs  should  like- 
wise increase  from  the  crown  to  the  springing.  Perronet  has  given  as 
a  rule  for  the  depth  at  the  crown,  the  formula:  d=  0,0694r+  0,325 
metres,  or,  in  English  measure  d  =  0,0694r  -f- 1  foot,  in  which  for- 
mula r  is  the  greatest  radius  of  curvature  of  the  intrados. 

For  arches  whose  radius  is  above  48  feet,  or  15  metres,  this  for- 
mula gives  greater  dimensions  than  is  given  in  ordinary  practice. 
The  depth  of  voussoirs  must  be  regulated  by  the  strength  of  the 
materials,  and  the  position  of  the  line  of-resistance  in  the  arch.  The 
joints  being  kept  very  thin,  so  that  the  mortar  serves  rather  to  dis- 
tribute the  pressure  uniformly  over  the  bed  of  the  stone,-  it  will  be 
found  that  a  thickness  which  reduces  the  strain  to  225  Ibs.  per 
square  inch  of  surface,  allows  of  ample  security  for  the  average  of 
materials.  One-half  this  thickness  must,  however,  exist  on  each  side 
of  the  line  of  resistance. 


34  TEST  OP  THE  EQUILIBRATION  OF  ARCHES. 

Remark  1.  225  Ibs.  per  square  inch  is  only  ^  of  the  absolute  strength  of  sound  sand- 
stone  and  limestone.  In  the  celebrated  bridge  at  Neuilly,  near  Paris,  built  in  1768  to 
1780,  by  Perronet,  the  estimated  pressure  per  square  inch  is  280  Ibs. 

Remark  2.  When,  as  in  the  sequel  we  always  do,  we  take  the  .thrust  or  pressure  at 
the  top  of  the  crown  of  the  arch,  and  in  like  manner,  only  consider  a  rotation  round  the 
lowest  point  of  the  angle  of  rupture,  it  is  the  more  necessary  to  assume  this  high  degree 
of  security,  and  to  give  the  arch  corresponding  depth  of  voussoirs,  as  in  these  assump- 
tions we  only  get  the  least  value  of  the  pressure.  Besides,  it  is  chiefly  the  upper  edges 
of  the  voussoirs  at  the  crown  of  the  arch,  and  the  lower  edges  of  those  at  the  joints  of 
rupture  that  have  to  withstand  the  pressure,  and,  therefore,  soonest  give  way;  the 
depth  we  have  indicated  on  each  side  the  line  of  pressure  is,  therefore,  necessary  to 
insure  stability. 

§  19.  Test  of  the  Equilibration  of  Arches. — The  investigation  of 
the  stability  of  an  arch  may  be  gone  through  as  follows:  let  JIB  CD, 
Fig.  24,  be  the  one-half  of  the  arch  to  be  examined,  and  CDHK  the 

Fig.  24. 


spandril  wall,  which  for  simplicity's  sake  we  shall  assume  to  be  of 
the  same  density  as  the  arch.  First,  divide  the  arch  in  any  conve- 
nient number,  in  this  case  six,  equal  or  unequal  parts,  by  lines  Ef^ 
E2Fy,  E3F3,  &c.,  in  the  direction  of  the  joints,  and  determine,  not 
only  the  area  and  the  centres  of 'gravity  Tv  T2,  T3 . .  .  of  these  parts, 
but  also  the  areas  and  centres  of  gravity  8V  S2,  S3 ...  of  the  super- 
incumbent parts  t\H,  F2LV  F3L2 . . .  This  done,  take  the  statical 
moment  of  the  first  part  AFl  and  F^H  referred  to  the  first  point  of 
division  E^  and  divide  their  sum  by  the  vertical  distance  of  this 
point  of  division  from  the  horizontal  DJV*  drawn  through  the  crown. 
In  like  manner  take  the  moments  of  AFV  EJ?2,  F^H  and  F2LV 
referred  to  the  second  point  of  division  Ez,  and  divide  the  sum  of 
these  moments  by  the  vertical  distance  of  this  second  point  from  the 
horizontal  DN.  Again,  determine  the  moments  of  the  parts  of  the 
arch  AFV  E^F»  E2F3,  and  the  parts  of  the  spandril  Fj#,  F2Lp  F3L2, 
referred  to  the  edge  E3,  and  divide  that  sum  by  the  vertical  distance 
of  the  point  -E3,  from  the  horizontal  DJV,  &c.  By  going  through 
this  process  for  all  the  parts,  from  Jl  to  B,  we  arrive  at  the  forces 
that  must  be  applied  at  D  to  prevent  rotation  round  the  points 


TEST  OF  THE  EQUILIBRATION  OF  ARCHES.  35 

Ev  E2,  E3,  «fec.,  and  the  greatest  of  these  forces  is  that  which  has  to 
be  taken  as  acting  at  the  crown. 

Having  done  this,  multiply  the  sum  of  the  areas  J1F1  +  F^H  by 
the  tang.  (a1  —  P),  and  again  AFl  +  E^  +  F^H  +  F2L:  by  tang. 
(«2 —  P),  &c.,  (where  av  aa . . .  are  the  several  angles  of  inclination  of 
the  joints  with  the  horizon),  and  find  the  greatest  value  of  these 
products.  If  the  greatest  of  these  values  be  less  than  that  "neces- 
sary to  prevent  rotation  round  Ev  E2,  E3 . . . ,  there  need  be  no 
further  consideration  of  these  forces ;  but  if  it  be  greater,  then  must 
this  value  be  introduced  as  the  pressure  in  the  crown,  and  not  that 
first  found. 

Lastly,  it  has  to  be  determined  whether  the  horizontal  force  so 
found  is  not  sufficient  to  dislocate  the  arch  by  pushing  or  turning 
out  a  part  of  it. 

With  the  horizontal  thrust,  determined  as  above,  we  can  examine, 
as  shown  in  §  16,  the  conditions  of  stability  of  the  abutment. 

Example.  The  relative  stability  of  the  arch  in  Fig.  24,  may  be  calculated  as  follows : 
area  of  the  part  J1F1  =  6,89  square  feet ;  area  of  the  piece  FJH  above  d^  8,48  square 
feet,  the  lever  of  the  former  referred  to  _E,  =2,50,  and  of  the  latter  =  2,45;  i.  e,  the 
moment  of  both  =  6,89  .  2,5  +  8,48  .  2,45  =  38,00] .  The  distance  of  E,  from  DN, 
or  leverage  of  horizontal  force  in  D  =  1.50  ;  and,  therefore,  the  first  value  of  this  force 

-p   —       ^ y  —  9.S.33  .  y  Ibs.     Area  of  second  part  -E,-F2  =  7il«>i  an(i  tne  Part  of  spandril 

above  it  F2Ll  =  11,02  square  feet;  the  moment  of  both  referred  to  E2=  17,52  +  23,69 
=  41, 21,  adding  to  this  the  moment  of  ^Z,  =  38+  15,37  .5.10  =  38  +  78,39=116,39, 
and  hence  the  moment  of  the  whole  piece  JlL2=  157,60  ;  the  distance  of  E2  from  DN 

=  2,35,  and  hence  the  second  value  of  the  horizontal  force  in  D  =  157-60  •  X  _  67  05  , 

*»  !i,35 

•y  Ibs.  Again,  the  area  of  the  third  piece  E1F3  =  7,68,  and  of  the  part  of  spandril  above 
it  F^L3^  16,51  square  i'eet;  the  moment  of  both ^46,61,  adding  to  this  the  moment 
of  the  piece  E,H=  1 57,60  +  16602=323,62;  we  find  the  moment  of  the  whole 
=  370,23 ;  and  as  the  distance  of  the  point  E3  from  HN=  3,90,  the  value  of  the  force  in 

T)  =        ' '-Z.  =:  94,93  .  y  Ibs.     Proceeding  in  this  manner,  a  value  of  the  force  that 

*4  3,90 

has  to  counteract  the  tendency  to  rotation  round  jE4  = \        *  —  *  18,97  .  y  Ibs.;  and 

a  fifth  force  in  reference  to  rotation  round  E5  = ! 1?  =  137,68  .  y  Ibs.;  and, 

lastly,  in  reference  to  rotation  round  B.  a  force  s=          ' !_?!  ^  157, 74.  y  Ibs.    As  this 

11,6 

is  the  greatest  value  found,  we  put  the  pressure  or  thrust  at  the  crown,  P=  151,74  .  y, 
or,  taking  the  weight  of  masonry  as  150  Ibs.  per  cubic  foot,P  =  22761  Ibs.  The  depth 
of  arch  at  crown  is  1 ,3  feet ;  and,  therefore,  the  area  for  each  foot  of  length  of  the  arch 
=  144  .  1,3  =  187,2  square  inches;  and  hence  the  pressure  on  each  square  inch 

22761 

=  122  Ibs.,  supposing  the  line  of  resistance  to  bisect  the  voussoirs. 

187,2 

If,  with  M.  Petit,  we  take  the  angle  of  repose  =  30°,  we  obtain  for  the  force  to  pre- 
vent dislocation  of  the  arch  by  sliding,  the  following  values.  The  joints  £,Fj,  EJfy, 
£3F3...are  inclined  to  the  horizon  at  the  angles  83°  40', 77°  20',  71°,  64°  407,  58°  207, 
52°,  respectively,  therefore, 

P,  =  (6,89  -f  8,48)  tang.  (83°  40'  —  30°)  .  y  =  15,37  .  tang.  53°  4tf  .  y  =  20,9  .  y  Ibs. ; 
P2  =  (15,37+18,17)  tang.  (77°  207  — 30°)  .y=  33,54  .  tang.  47°  207  .  y  =  36,4  .y  Ibs.; 
P3  =  57,73  .  tang.  41°  .  y  =  50,1  .  y  Ibs. 
P4  =  90,56  tang.  34°  40'  .  y  =  62,6  .  y  Ibs. 
P5=  134,13  tang.  28°  20'  .  y=  72,3  .  y  Ibs. 
P6  =  188,53  tang.  22°  .  y  =  76,2  .  y  Ibs. 


36  TABLES  FOR  ARCHES. 

and,  therefore,  the  greatest  horizontal  pressure  counteracting  sliding  =  76,2  .  y  Ibs.  As, 
however,  this  pressure,  in  its  tendency  to  cause  rotation  round  an  inner  joint,  amounts 
to  151,7  .  y.  it  is  evident  that  a  sliding  cannot  take  place.  And  in  like  manner  it  is 
easy  to  convince  ourselves  that  neither  rotation  nor  sliding  outwards  is  possible.  As  to 
the  stability  of  the  abutment  OUK,  the  moment  of  the  force  P  referred  to  0  as  an  axis 
of  rotation  =  151,74  .  y  .  OV=  151,74  .  18  .y  =  2731  .y  Ibs.;  the  moment  of  the  loaded 
arch  JiBKH,  is : 

1760,2  .  y-\-  188,53  .  Uu .  y=(1760,2+  188,53  .  6,8)  y  =  3042  .  y  Ibs., 
and  that  of  the  pier  =  343  .  y  Ibs.;  hence  the  moment  resisting  rotation  round 
O  =  (3042  -f  343)  .  >•=  3385  .  y  Ibs.,  and,  therefore,  heeling  cannot  possibly  take  place. 
If,  however,  more  ample  security  be  desired,  we  must  substitute  for  P,  1,9  P,  as  above 
explained,  and,  therefore,  take  the  moment  of  the  force  to  produce  heeling  =  5189  .  y, 
and  thus  we  see  that  our  abutment  would  be  too  thin ;  instead  of  5,45  feet  thickness, 
it  would  require  from  II  to  12  feet.  For  a  thickness  of  11  feet,  the  moment  of  sta- 
bility =  1760,2  .  y-f  188,53  .  11  y+  1281  .  y  =  5JJ5  .  y,  which  would  prove  a  sufficient 
stability. 

§  20.   Tables  for  Arches. — In  order  to  facilitate  investigations  on 
the  stability  of  arches  of  the  more  usual  forms, 
Fig.  25.  ]yj  Petit  calculated  a  series  of  tables  of  which 

we  shall  here  give  a  short  abstract.  The  first 
of  these  tables  refers  to  semi-circular  vaults,  as 
in  Fig.  25,  the  second  refers  to  semi-circular 
arches  with  spandril-walls  at  an  angle  of  45°  as 
shown  by  the  dotted  line  in  Fig.  23.  The  third 
table  refers  to  semi-circular  arches  with  hori- 
zontal spandrils,  as  shown  by  the  dotted  line  in 
Fig.  22,  and  the  fourth  table  refers  to  segmental 
arched  vaults. 

In  the  first  three  tables,  the  two  first  vertical  columns  contain 
the  proportions  of  the  arches ;  the  third  column  contains  the  angle 
of  rupture;  the  fourth  and  fifth, co-efficients  of  horizontal  thrust,  in 
terms  of  the  radius  or  half  span,  and  the  weight  of  the  materials 
(see  example  1  following);  and  in  the  sixth,  the  co-efficient  of  the 
maximum  thickness  of  abutment  in  terms  of  the  half  span. 

To  apply  these  tables,  we  have  to  look  in  column  1  for  the  ratio 

k  =  -2.  of  the  radius  of  the  extrados  to  that  of  the  intrados,  and 
r1 

pass  along  horizontally  to  the  fourth  and  fifth  columns,  and  the 
greater  of  the  numbers  found  in  these  two  columns  is  to  be  taken  as 
a  co-efficient  by  which  to  multiply  the  square  of  rv  the  radius  of 
intrados,  and  the  weight  per  cubic  foot  y  of  the  masonry,  the  pro- 
duct of  which  gives  the  horizontal  thrust  in  question.  The  sixth 
column  gives  the  thickness  of  abutment,  supposing  the  height  infi- 
nite, by  multiplying  the  co-efficients  there  found  by  the  radius  rr  - 
For  low  abutments,  the  thickness  is  less,  and  should  be  calculated 
according  to  §  17. 

The  fourth  table  contains  in  its  first  column  the  ratio  Jc  =  ^, 

and  in  the  other  columns  the  thrust  of  the  arch  for  various  propor- 
tions of  the  span  s  to  the  versed  sine  or  height  h.  This  latter  table 
is  only  applicable  when  the  angle  of  rupture,  given  in  the  first  table,  is 
less  than  the  half  of  the  central  angle  a,  and  of  the  arc  of  the  vault. 


TABLES  FOR  ARCHES. 


37 


TABLE  I. 
SEMICIRCULAR  ARCH  WITH  PARALLEL  VAULTED  SURFACES. 


Ratio  of  the 
radii. 

*-r7 

Ratio  of  radius 
of  intrados  to 
depth  of  vous 
soir. 

Angle  of 
rupture. 

Co-efficient  p  of  the 
thrust  of  arch  ; 

Co-efficient    for 
greatest  thick- 
ness  of  abut 
ments. 

for  rotation. 

for  sliding. 

2,732 

1,154 

0°  00' 

0,00000 

0,98923 

2,70 

1,176 

13    42 

0,00211 

0,96262 

2,50 

1,333 

35     52 

0,02283 

0,80346 

2,20 

1,666 

51       4 

0,08648 

0,58767 

2,00 

2,000 

57     17 

0,13017 

0,45912 

1,3223 

1,80 

2,500 

61     24 

0,16373 

0,34281 

1,1414 

1,60 

3,333 

63    49 

0,17517 

0,23874 

0,9525 

1,55 

3,636 

64       3 

0,17478 

0,21464 

0,9031 

1,50 

4,000 

64       9 

0,17254 

0,19130 

0,8527 

1,45               4,444 

64      5 

0,16798 

0,16872 

0,8007 

1,40               5,000 

63    48 

0,16167 

0,14691 

0,7838 

1,35              5,714 

63     19 

0,15287 

0,12587 

0,7622 

1,30               6,666 

62     14 

0,14330 

0,10559 

0,7370 

1,25              8,000 

61     15 

0,12847 

0,08608 

0,6987 

1,20             10,000 

59     41 

0,11140 

0,06733 

0,6504 

1,15             13,333 

57       1 

0,09176 

0,04935 

0,5905 

1,10             20,000 

53     15 

0,06754 

0,03213 

0,5066 

1,05            40,000 

46     22 

0,03813 

0,01568 

1,02           100,000 

38     12 

0,01691 

0,00618 

1,00                oo 

0     00 

0,00000 

0,00000 

TABLE  II. 
SEMICIRCULAR  ARCHES,  MASONRY  AT  THE  BACK,  OF  45°  INCLINATION. 


Ratio  of  the 

Ratio  of  radius 

Co-efficient  p  of  the 

Coefficient    for 

radii. 

of  intrados  to 

Angle  of 

thrust  of  arch  ; 

greatest  thick- 

7          r* 

depth  of  vous- 

rupture. 

ness  of  abut- 

K =  _. 

r, 

soir. 

for  rotation,  i  for  sliding. 

ments. 

2,00 

2,000 

60° 

0,26424    0,74361 

1,7264 

1,80 

2,500 

60 

0,29907 

0,57383 

1,5147 

1,60 

3,333 

60 

0,31245    0,42191 

1,2990 

1,55 

3,636 

61 

0,31222    0,38673 

1,2437 

1,50 

4,000 

61 

0,30996    0,35266 

1,1877 

1,45 

4,444 

60 

0,30587  i  0,31971 

1,1308 

1,40 

5,000 

59 

0,30001 

0,28787 

1,0954 

1,35 

5,714 

58 

0,29285 

1,0823 

1,30 

6,666 

57 

0,28231 

0,22756 

1,0626 

1,25 

8,000 

54 

0,27102 

1,0412 

1,20 

10,000 

50 

0,25806 

0,17171 

1,0160 

1,15 

13,333 

47 

0,24477 

0,9894 

1,10 

20,000 

42 

0,23292 

0,12032 

0,9652 

1,05 

40,000 

36 

0,22902 

0,9571 

VOL.  II  —4 


TABLES  FOR  ARCHES. 


TABLE  III. 
SEMICIRCULAR  ARCHES,  WITH  HORIZONTAL  MASONRY  ABOVE. 


Ratio  of  the 
radii 

ft_& 

rt 

Ratio  of  radius 
of  intrados  to 
depth  of  vous- 
soir. 

Angle  of 
rupture. 

Co-efficient  p  of  the 
thrust  of  arch  ; 

Co-efficient    for 
greatest  thick- 
ness  of  abut- 
ments. 

for  rotation. 

for  sliding. 

2,00 

2,000 

36°       0,05486 

0,50358 

1,3834 

1,80 

2,500 

44 

0,08508 

0,37901 

1,2001 

1,60 

3,333 

52 

0,12300 

0,26755 

1,0082 

1,55 

3,636 

54 

0,13027 

0,24173 

0,9584 

1,50 

4,000 

56 

0,13648 

0,21673 

0,9075 

1,45 

4,444 

57 

0,14122 

0,19256 

0,8554 

1,40 

5,000 

59 

0,14421 

0,16920 

0,8018 

1,35 

5,714 

60 

0,14504 

0,14666 

0,7465 

1,30 

6,666 

61 

0,14332 

0,12495 

0,7379 

1,25 

8,000 

62 

0,13872 

0,10405 

0,7260 

1,20 

10,000 

63 

0,13073 

0,08397 

0,7048 

1,15 

13,333 

64 

0,11895 

0,06471 

0,6723 

1,10 

20,000 

65 

0,10279 

0,04627 

0,6249 

1,05 

40,000 

69 

0,081755 

0,02865 

0,5573 

1,00               oo 

75 

0,055472 

0,01185 

TABLE  IV. 
VAULTED  ARCHES,  WITH  PARALLEL  ARCHED  SURFACES. 


Ratio  of 
the  radii 


Co-efficient  p  of  the  thrust  of  the  arch. 


k=r\ 

n 

(  =  4  h 

t=5h 

*=6  h 

*=7  h 

s=8h 

»=10A 

*  =  16  A 

1,40 
1,35 

1   5*0 

0,15445 
0,14771 

n  1  37«/L 

0,14691 
0,13030 

0  193Q1 

0,14691 

0,12587 
n  ift«fto 

0,146910,146910,14478 
0,12587  0,12587  0,12405 
n  i  ri^Q  n  1  ftf^o  n  i  n.in« 

1,25  0,12547!0,11402  0,1000910,08668  0,08608  0,08483  0,07180 

1,20  0,11023  0,10196  0,09102*0,07999  0,06981 0,06636  0,05616 

1,15  iO,09123  0,08634  0,07866!o,07050  0,06259  0,04904  0,04116 

1,10  0,06737  0,06563  0,06158!0,05666  0,05160  0,02414  0,02681 

1,05  ,0,03776  0,03804  0,03709  0,03550  0,03357  0,02944  0,01882 

1,01  jO,00834iO,00871  0,00886,0,00889  0,00885  0,00862  0,00747 


TABLES  FOR  ARCHES. 


39 


The  following  table  contains  a  synopsis  of  the  relative  dimensions 
of  segmental  arches. 


Ratio  of  the  span 
to  height 
1 

Half  central 
angle  a. 

sin.  a,. 

Ratio  of  radius  of 
intrados  r,  to 
height  A 

r 

!i 

h 

4 

53°     r    30" 

0,8000 

2,500 

5 

43     36     10 

0,6897 

3,625 

6 

36     52     10 

0,6000 

5,000 

7 

31     53     26 

0,5283 

6,625 

8 

28      4     20 

0,4706 

8,500 

10 

22     37     10 

0,3846 

13,000 

16 

14     15      0 

0,2462 

32,500 

Example  1.  A  semicircular  arch  with  horizontal  road-way  over  it,  having  radius  of 
intrados  rl  =  10  feet.  What  should  be  the  dimensions?  What  will  be  the  thrust? 
According  to  Perronet's  formula,  d=  0.0694  .  10+  1  =  1,694  feet,  for  which  take  1,7  j>,  •> 

feet.  We  have  now  r,=  11,7  and  k  =  -1  =  1,17.  From  Table  3,  the  angle  of  rup- 
ture is  63|°,  the  co-efficient  of  horizontal  thrust  =  0,1 190+  f  .  0,0118  =0,1237;(0,0118 
being  the  difference  between  .119,  and  the  number  next  above  it).  Taking  150  Ibs.  per 
cubic  foot  as  weight  of  masonry,  the  thrust  at  crown  =  0,1237  .  150  .  10a=1855  Ibs. 
For  the  extreme  thickness  of  abutment,  we  have  from  the  same  table  the  co-efficient 
0,6723+ f  .0,0325=0,6855,  and,  therefore,  the  thickness  =  0,6855  .  10  =  6,85  feet. 
For  low  abutments,  the  formula  of  §  17  gives  smaller  dimensions. 

Exampk  2.  What  dimensions  and  forces  correspond  to  a  vault  of  10  feet  span,  and  2 
feet  rise?  Here  we  have  _  =  £,  therefore,  the  half  central  angle  a  =  43°  36'  10",  and 

sin.  »  =  0,6897,  and  the  radius  r  =  3,625  .  2  =  7,25  feet.     Table  4  gives  the  co-efficient       k.,  3 
of  horizontal  thrust,  (as  s=5  h,  and  according  to  Perronet's  formula:  d=  1,5,  so  that 
&  =  l,2);b=  0,1 0196,  and  hence  the  thrust  =  0,102  .  150  .  7,25a  =  804  lbs.=  ^vy 

Remark  1.  That  the  part  of  the  abutment  on  which  the  arch  rests  may  not  be  thrust 
away,  it  is  necessary  that  the  horizontal  thrust  P=pr*y  should  be  less  than  £/«  (r,0 — r?) 
y  the  friction  on  the  bed.  If  this  be  not  the  case,  as,  for  example,  in  very  flat  arches, 
this  sliding  out  of  the  upper  part  of  the  abutment  must  be  prevented  by  artifices,  such  as 
iron  tie  rods.  The  co-efficient  of  friction  /=0,76,  therefore,  £/=  0,38,  and,  therefore, 
the  strength  of  the  ties  must  be  such  as  to  resist  a  force  =p  —  0,38  a.  (k2 — 1)  r,3  y. 
This  is  the  state  of  the  case  when  s  =  4  h  and  k  is  less  than  1,06  ;  when  *=  5  h  to  10 
h,  and  k  less  than  1,15,  and  when  s=  16  A,  this  sliding  is  sure  to  take  place. 

Remark  2.  The  literature  on  the  subject  of  arches  is  very  extensive ;  but  the  theories 
treated  therein  are  not  always  admissible,  because  the  assumptions  are  inconsistent  with 
experience.  We  shall  here  only  mention  the  authors  whose  theories  and  investigations 
are  generally  accepted  as  the  best  approximations.  We  refer,  therefore,  to  Coulomb, 
"  Theorie  des  machines  simples,"  who  first  gave  a  rational  theory  of  the  arch,  and  such 
as  is  in  substance  given  in  the  foregoing  paragraphs.  This  theory  is  given  with  greater 
completeness  by  Navier,  "  Resume  des  Lecons  sur  1'application  de  la  Mecanique,"  t.  i. 
There  are  papers  by  Audoy,  Garidel,  Poncelet  and  Petit,  in  the  "  Memorial  de  1'officier 
du  genie."  The  substance  of  the  papers  of  Garidel  and  Petit,  and  their  tables,  are  given 
by  Mr.  Hann  in  his  Treatise  on  Bridges,  published  by  Weale,  1839.  Moseley's  paper 
on  the  "Theory  of  the  Arch,"  is,  perhaps,  the  most  elegant  exposition  of  this  interesting 
and  important  subject.  The  works  of  Robison,  Whewell,  Eytelwein,  Gerstner,  and 
others,  contain  particular  expositions  of  Coulomb's  theory.  Hagen  has  published  an 
interesting  essay,  entitled  "  Uber  Form  und  Stilrke  gewolbter  Bogen,"  Berlin,  1844. 


40 


WOODEN  STRUCTURES. 


CHAPTER   III. 

THEORY  OF  FRAMINGS  OF  WOOD  AND  IRON. 

§  21.  Wooden  Structures. — Structures  of  wood  and  of  iron  differ 
essentially  from  those  in  stone,  in  that  these  materials  are  subjected 
to  what  have  been  termed  tensile  and  transverse,  as  well  as  com- 
pressive  strains,  to  which  latter  alone  masonry  is  exposed.  Hence, 
in  carpentry  and  iron-work,  the  pieces  of  which  the  framings  are 
composed  are  not  only  laid  one  upon  the  other,  but  are  morticed, 
tenoned,  fished,  bolted,  strapped,  &c.,  to  unite  them  together.  The 
principal  axis  of  the  pieces  of  any  framing  may  be  horizontal,  in- 
clined, or  vertical.  In  the  first  case,  they  are  termed  beams  or  joists; 
in  the  second,  rafters,  braces,  or  spears,  &c.;  in  the  other,  posts, 
pillars,  uprights,  &c.  According  to  the  function  they  fulfil,  some 
pieces  are  termed  struts  or  spears  (viz:  those  resisting  compression), 
and  others,  ties  or  braces  (i.  e.  those  resisting  tension). 

To  investigate  the  stability  or  equilibrium  of  a  framing,  it  is  essen- 
tial, in  the  first  place,  to  know  the  forces  and  weights  which  the 
framing  has  to  counteract.  From  these  we  determine,  not  only  the 
forces  which  individual  pieces  have  to  withstand,  but  the  forces  act- 
ing at  the  points  of  connection,  and  the  strains  or  pressures  upon 
the  points  of  support.  Each  part  should  have  such  form,  position 
and  dimensions,  as  to  completely  withstand  every  force  acting  on  it. 
As  to  the  connection  of  the  pieces  of  a  framing  with  each  other, 
we  have  principally  to  distinguish  bolts  and  pins,  tenons  and  mor- 
tices, scarfs  and  shoulders.  Bolts  and  pins  counteract,  or  take  up 
all  forces  passing  through  their  axes.  Tenons  and  mortices  counter- 
act only  forces  acting  in  certain  directions,  and  shoulders  or  scarfs 
counteract  such  forces  as  are  directed  at  right  angles  to  the  plane 
of  the  shoulder. 

§  21*.  A  beam  AB,  Fig.  26,  lying  on  inclined  planes,  is  in  an 
instable  condition,  unless  friction  or  some 
artificial  fastening,  as  bolts  or  mortices  re- 
tain it.  To  establish  equilibrium,  it  is  a 
necessary  condition  that  the  vertical  SG 
passing  through  the  centre  of  gravity  of 
the  beam,  should  pass  through  the  point 
C,  in  which  the  normals  to  the  ends  A 
and  B  of  the  planes  intersect  each  other, 
for  only  then  are  the  two  components  N 
and  P,  into  which  the  weight  G  of  the 
beam  may  be  decomposed,  taken  up  or 


Fig.  26. 


WOODEN  STRUCTURES.  41 

counteracted  by  the  planes.     If  <*  and  ft  be  the  angle  AOK  and 
BOL  of  the  planes  to  the  horizon,  these  forces  are: 

,,  ~         G  sin.  ft  iD          G  sin.  a 

Jv  = ,  and  P 


(sin.  a  +  ft)  sin.  (a  +  ft) 

If,  again,  I  be  the  length  JIB  of  the  beam,  «  the  distance  AS  of 
its  centre  of  gravity  S  from  the  end  A,  and  8  the  angle  of  inclina- 
tion BAM  of  the  beam  to  the  horizon,  then  the  horizontal  projection 
of  AS  =  s  is  AM  =  s  cos.  8,  or  =  AC  sin.  a,  but  as 

Jic  _  AB  sin.  ABC  _  I  sin.  (90°  —  £  +  6)  _  I  cos,  (ft  —  s) 
sin.  A  CB  sin.  (a,  +  ft)  '    sin.  (*  +  ft)  ' 

l  *in'  a  C°8'  ^  ~  *     and,  therefore,  we  have  the 


sin.  (o  +  ft) 
equation  of  condition  : 

s.  sin.  (a  -\-  ft)  cos.  8  =  I  sin.  a  cos.  (ft  —  8). 

If  one  of  the  planes  be  horizontal  as  AO  in  Fig.  27,  then  as  a  =  0, 
we  have  s  sin.  ft  cos.  8=0,  i.  e.  ft  =  0,  or  the  other  plane  must  like- 
wise be  horizontal.  In  order  to  prevent  slipping  of  the  beam  in 
every  other  position,  we  must,  Fig.  28,  mortice  one  end  of  the  beam 

Fig.  27.  Fig.  28. 


as  A,  or  fasten  it  in  some  way.  The  pressure  which  the  end  of 
the  beam  there  exerts  on  the  inclined  plane  OB  may  be  deduced 
from  the  theory  of  the  bent  lever  MAC,  whose  arm  AM  =  AS  cos. 
SAM=  s  cos.  5,  and  AC  =  AB  cos.  BAG  =  /  cos.  (ft  —  5),  and 
hence  P  the  pressure  required 

G-  s  cos.  S  i 

~  I  cos.  (ft  —  8) 

As  the  pressure  on  the  point  of  support  A  is  equal  to  the  mean 
of  all  the  forces  acting  on  AB,  we  may  assume  that  the  vertical 
pressure  Gl  =  G,  and  its  counter  pressure  Pl  =  P,  acts  at  this 
point.  If,  therefore,  we  decompose  this  latter  into  the  horizontal 
force  #j  ==  Pl  sin.  ft,  and  the  vertical  force  Vl  —  P1  cos.  ft,  we  ob- 
tain for  the  total  pressure  in  A  the  horizontal  component  or  thrust 

H.  =  G  s  sm"  ft  cos"  8,  and  the  vertical  component,  or  vertical  pres- 

l  Cos,  (ft  — 8)    ' 

sure: 

4* 


42 


THRUST  OF  ROOFS. 


from  which  we  can  easily  calculate  the  magnitude  and  direction  of 
the  total  pressure  or  strain. 

For  the  case  of  a  beam  leaning  on  a  wall,  Fig.  29,  ft  =  90°, 
hence : 


Fig.  30. 


I  sin.  6 
the  weight  of  the  beam. 

Fi?.  29. 


For  the  case  of  a  beam  leaning  on  a  wall  inclined  at  the  same 
angles  as  the  beam,  as  in  Fig.  30  at  .B,  &  =  6,  hence: 

P  =  G  -  cos.  «, 

Jf=  G  S-  sin.  6  cos.  «,  and  V=  G  /I  _  -  cos.  62V 

§  22.  Thrust  of  Roofs. — The  formulas  found  in  the  preceding  para- 
graphs are  immediately  applicable  to  calculating  the  thrust  of  rafters 
or  "couples"  for  roofs  (Fr.  fermes}.  According  to  these,  we  have  in 
the  case  of  simple  lean-to  and  coupled  roofs,  as  in  Figs.  31  and  32, 


Fig.  31. 


Fig.  32. 


for  the  horizontal  thrust  acting  at  the  lower  and  upper  end : 
H  =  —  cotg.  S,  or,  as  in  this  case  8=%l,H=^G  cotg.  6 ; 
again  the  vertical  pressure  at  the  upper  end=  0,  and=  G  the  weight 


THRUST  OF  ROOFS. 


43 


of  the  couple  and  its  load  at  the  lower  end.  If  we  put  the  height 
of  roof  BC  =  h,  and  the  span  or  width  AC  =  DC  =  b,  then 

cotg.  8  =  -,  and  hence  theitfirust  of  the  couple  =  ^  G  _;  and  thus 

h  h 

we  see  that  the  horizontal  thrust  increases  directly  as  the  span,  and 
inversely  as  the  height  or  pitch  of  the  roof.  The  usual  limits  of  h 
are  between  2  b  and  \  b.  The  former  ratio  is  that  of  church  roofs 
of  the  Saxon  and  Norman  period,  and  the  latter  that  of  the  flat 
Italian  roofs  of  modern  houses.  In  the  former,  8  =  26°  34',  and 
in  the  latter  63°  26'.  The  thrust  of  the  couples  is  very  great  in  flat 
roofs ;  in  the  Italian  roof,  for  instance,  as  above  specified,  the  thrust 
equals  the  whole  weight  of  the  couple  and  load ;  in  the  Saxon  roof 
the  thrust  is  not  above  one-fourth  of  this.  The  feet  of  the  couple 
must  be  morticed,  or  otherwise  fastened  into  the  beam  (tie-beam)  to 
prevent  sliding.  The  entire  pressure  of  a  rafter  at  its  foot  A  is : 


and  for  BJ1H=  $,  the  angle  made  by  the  line  of  pressure  with  the 
horizon,  we  have 

G  G          2A       0  , 

tang.  $  =  —  = =  —  =2  tang.  8. 

H       ,   ^,  b        b 


Thus  we  may  find  the  direction  of  the  total  thrust  at  the  foot,  b 
doubling  the  height  of  the  couple ;  or,  by  making  CE  =  2  .  C 
and  drawing  a  line  through  the  foot  Jl,  from  the  point  E,  and  pro- 
ducing it  to  R. 

For  the  pair  of  rafters,  Fig.  32,  in  which  the  rafters  are  of  equal 
length,  these  exert  on  each  other  only  a  horizontal  thrust ;  but  if 
the  rafters  be  of  unequal  length,  as  in  Fig.  33,  the  force  P  with 
which  one  rafter  presses  upon 

the  other,  deviates  by  a  certain  Fig.  33- 

angle  from  the  horizontal.  If 
G  be  the  weight  of  one  rafter 
AB,  and  Gl  that  of  the  other 
CB,  and  if  8  and  ^  be  the 
angle  of  inclination  of  these 
rafters  to  the  horizon,  and  if  ft 
be  the  angle  of  inclination 
BDC  of  the  plane  in  which  we 
may  conceive  the  rafters  to 
abut  on  each  other,  and  against 
which  the  force  P  acts  at  right 
angles,  we  have  :  '  k-^i -•('). 

p       i      G  cos.  8          n       ,              G.  cos.  8, 
P  =  * ,  and  =  i l- l- 


cos.  (180°  —  0— 


-,  hence 


—  G  cos.  8  cos.  (ft  +  8,)  =  G,  cos.  8X  cos.  (ft  —  8),  or 
G  (sin.  ft  sin.  8,  —  cos.  ft  cos.  «,)  _  Gl  (sin.  ft  sin.  8  -f  cos,  ft  cos.  8)  t 


sin.  ft  cos. 


sin.  ft  cos.  8 


44 


COMPOUND  ROOFS. 


dividing,  we  have : 

G  (tang.  8j  —  cotg.  ft]  =  Gj  tang.  8  4-  cotg.  ft),  thus 
cotang.  ft  =  G  tang.  1,-G,  tang.  *  ^ 

G  +  Gj 
And  from  this  we  have  the  horizontal  thrust  of  both  rafters  : 

H=s  p  8in.fi=i  Gsin.ftcos^_= ^G 

cos.  (ft  —  8)          cotg.  ft  +  tang.  8 


tang.  S  +  tang.  8X 

As  to  the  vertical  pressures  V  and  Vl  at  the  rafter  feet,  the  one 
is  equal  to  the  weight  G,  minus  the  vertical  component  Q  =  P 
cos.  ft,  and  the  other  is  equal  to  the  weight  G2  plus  this  compo- 
nent ;  or, 

V=  G- 


tang.  8  +  tang.  ^ 

and  V,  =  Gj  +  J  (G  tang.  S.-G,  tang. 
\      tang.  8  +  tang.  8X 

Example.  The  roof  JIBD  (Fig.  32),  is  40  feet  span,  and  30  feet  height,  and  consists 
of  couples  4  feet  from  centre  to  centre,  6  X  8  inch  scantlings  —  required  the  thrust. 
Assuming  each  square  foot  of  roofing  to  weigh  15  Ibs.,  we  have  for  the  load  on  each 
rafter  15x4  %/201^f~30T=  60  ^1300  =  2 163  Ibs.  The  rafter  itself  weighs  £  X  $  X  44 
^20" _j_  30"  =  45*  ^1300  =  529  Ibs.,  and,  therefore,  the  vertical  pressure  of  a  rafter 


V=  G  =  2163+  529  =  2692  Ibs.,  and  the  thrust  =  £  G 


.  2692  f-g-  =  897  Ibs. 


§  23.  Compound  Roofs. — In  many  framings,  as  in  mansard  roofs, 
the  rafter  DE,  Fig.  34,  does  not  rest  on  a  tie- 
Fig.  34.  beam,  but  on  a  second  rafter  CD,  and  this  again 
on  a  third,  and  fourth,  and  so  on.  That  the 
pressure  of  one  beam  may  be  completely  trans- 
ferred to  the  next  in  this  case,  it  is  necessary 
that  they  should  have  certain  relative  positions. 
These  positions  are  determined  by  the  condi- 
tions that  any  two  beams  abutting  against  each 
other  should  undergo  equal  horizontal  pressures. 
The  horizontal  pressure  of  the  rafter  DE,  is 
H  =  \  G  cotg.  8,  when  G  =  the  weight,  and  8 
its  inclination.  For  the  second  beam  or  rafter 

:'      ,  when  G:  and  S1  de- 


DC:H 


tang.  St  —  tang.  8 
note  weight  and  inclination  of  this  second  beam.    Hence  by  equating 
the  two  values,  we  have  : 

G+G, 


tang.  *, 


G  cotg.  8 

tang.  8X  —  tang.  8' 

=  tang.  8  +  i — ^—^  tang.  8  =  /2  +  _'\  tang.  8; 


and,  in  like  manner,  for  the  inclination  82  of  a  third  beam,  seeing 
that  the  horizontal  thrust  is  everywhere  the  same. 


SUPPORTED  RAFTERS. 


45 


G  cotg.  8 


tang.  82  —  tang 

ri    .if* 

tang.  82  =  tang.  8l  -\ L^ — ?  tang 

(JT 


,  hence 


and  in  like  manner  for  a  fourth: 

tang.  82  =  tang.  82  +  ^^_°J 
G,       G, 


If  each  beam  be  of  the  same  weight  G,  then 


tang. 


5  tang.  8,  tang.  83 


3  tang.  8,  tang, 
tang.  S4  =  9  tang.  8,  &c. 

If,  therefore,  in  this  form  of  roof,  the  height  EH,  Fig.  34,  corre- 
sponding to  the  first  beam  or  rafter  DE,  be  set  off  upwards  repeatedly, 
and  through  the  divisions  1,  3,  5,  7,  &c.,  lines  Dl,  D3,  D5,  D7,  &c., 
be  drawn,  these  lines  give  the  inclinations  of  the  other  rafters.  It 
is  also  evident,  that  the  figure  of  this  combination  of  rafters  is  that 
of  a  funicular  polygon  formed  by  the  weights  Gv  G2,  G3,  &c.  (see 
Vol.  I.  §  144),  and  this  coincidence  is  quite  explained,  if  we  conceive 
the  two  halves  of  the  weight  G  of  each  beam  collected  at  its  ends 
D,  C,  B,  A,  &c.,  and  pulling  downwards,  that  is,  if  we  assume  the 
weight  G  acting  at  each  of  these  points. 

If  we  take  the  beams  very  short,  and  very  numerous,  the  axis  of 
such  a  framing  becomes  a  catenary. 

§  24.  Supported  Rafters. — If  the  head  of  a  rafter  rests  on  a  pillar 
BC,  Fig.  35,  the  thrust  of  the  rafter  is  less 
than  when  it  merely  leans  on  a  vertical  wall.  Fig.  35. 

In  this  case,  according  to  §  21*,  the  pressure 
on  the  head  of  this  pillar  is: 

P=G-  cos.  8  =  i  G  cos.  8, 

and  the  horizontal  thrust: 
H=  P  sin.  8  =  i  G  cos.  8  sin.  8  =  £  G  sin.  2  8. 
As  the  pillar  supports  a  part  of  the  weight 
G  =  V  =  P  cos.  8  =  J  G  (cos.  S)2,  the  beam 
does  not,  of  course,  press  with  its  whole  weight 
G  on  the  foot  Jl\  but  with  a  force: 
F  =  G  —  i  G  (cos.  a)2  =  G  [1  —  i  (cos.  5)2]  =  J  G  [1  +  (sin.  «)*]. 
From  this  vertical  pressure,  and  the  horizontal  thrust  H,  we  get 
the  angle  <j>,  which  the  resultant  R  makes  with  the  horizon,  viz: 


Vl  1  +  (sin.  8)2 

If  we  introduce  the  depth  JiC—l  and  height  BC  =  h,  we  get 

H  =  — —  .  _,  while  in  the  case  of  the  beam  simply  leaning,  we 


46 


KING-POSTS. 


T  X-X 

had  H  =  -  .  _ .     If  each  unit  of  length  of  the  rafter  bears  a  load 

whose  weight  is  y,  we  have  G  =  </b2  +  h2 .  y,  and  therefore  in  the 

b  </b2  +  A2 


one  case  H  = 


and  for  the  other  H  = 


ported  at  the  ridge  by  i 
tral  wall  or  column.   Tl 


2  x/62  +  A*' 

so  that  if  the  pillar  support  the  rafter,  the  horizontal  thrust  is  so 
much  the  less  the  lower  the  roof;  while  for  roofs  without  such  sup- 
port, the  thrust  is  greater  as  the  roof  is  lower. 

That  the  post  BC  may  not  be  overturned  by  the  horizontal  force 
H,  it  is  necessary  to  support  it  by  a  wall. 

The  relations  of  the  forces 
now  discussed,  occur  in  the 
coupled  roof,  shown  in  Fig. 
36,  applicable  in  some  cases, 
where  the  rafters  are  sup- 

^a  cen- 
he  pil- 
lar takes  up  the  weights  J  G 
(cos.  «)2,  J  G  (cos.  6)2,  and 
transfers,  therefore,  the  ver- 
tical pressure  G  (cos.  s)2  to 
its  support,   and  the   hori- 
zontal thrust  H  =  ^  G  sin.  2  5.     There  is  no  side  support  required 
for  the  pillar,  as  the  horizontal  thrust  is  equal  on  each  side. 

Example.  For  the  roof  in  the  example  to  §  22,  the  loading  of  one  rafter  G  =  2692  Ibs., 
b  =  20  feet,  h  =  30  feet,  therefore,  tang.  >  =  |,  or  *  =  56°  18'  36";  and,  therefore, 
when  a  pillar  is  put  in,  the  horizontal  thrust  is : 

H=^lsin.  112°  37'  12"  =  673  tin.  67°  22'  48"  =  621  Ibs. 
4 

The  vertical  pressure  taken  up  by  the  pillar  is  F=  H^?  (cos.  56°  18'  36")2  =  746,3 
Ibs.;  and,  therefore,  the  beam  supports  a  strain  of  only  2692  —  746,3  =  1945  Ibs. 

§  25.  King-posts. — "Whilst  in  the  cases  just  considered  the  posts 
relieve  the  tie-beam  (or  walls  in  the  absence  of  a  tie)  of  a  part  of  the 
thrust  of  the  rafters,  the  king-post,  BC,  Fig.  37,  acts  in  a  very 

Fig.  37. 


KING-POSTS. 


47 


different  way ;  it  carries  a  part  of  the  weight  of  the  tie-beam  JlD, 
and  transfers  it  through  the  rafters  JIB  and  DB  to  act  as  thrust  on 
the  side  walls,  or  rather  as  tensile  strain  on  the  tie.  The  force  Q 
acting  through  the  king-post,  may  be  deduced  from  the  scantlings 
of,  and  kind  of  load  acting  on  the  beam  J1D.  If  the  load  be  uni- 
formly distributed,  it  may  be  assumed,  that  the  one  half  is  supported 
by  the  side  walls,  the  other  half  hangs  on  the  king-post ;  but  if  the 
load  be  applied  at  the  centre  of  the  tie,  it  must  be  considered  as 
acting  entirely  on  the  king-post.  The  force  Q  on  the  king-post  is 
decomposed  into  two  others  in  the  direction  of  the  rafters,  the  value 

of  each  of  which  is  S  =  — — — ;  and  if  we  combine  these  forces  with 
2  sin.  S 

those  arising  from  the  weights  G,  G  of  the  rafters,  we  get  the  hori- 
zontal thrust  in  A  and  D : 

H=  I  G  cotff.  «  +  Scos.  S  ==       "^  ^  .  cotff.  6, 
and  the  vertical  pressure  at  that  point : 
V=  G  +  Ssin.  8  =  G+^. 

For  bridges  and  roofs  of  great  span,  more  complicated  framings, 
with  two  or  more  posts,  and  termed  trusses,  are  applied.     Fig.  38 

Fig.  38. 


represents  a  truss  with  two  posts,  termed  queen-posts,  BC  and  B1C1, 
with  a  collar  beam  between  them  BBr  The  manner  of  calculating 
the  strains  in  this  framing  is  exactly  similar  to  that  for  the  simple 
couple  with  king-post.  From  the  load  on  a  queen-post  Q,  the  hori- 
zontal thrust  on  the  collar-beam  tending  to  compress  it,  and  acting 
on  the  side  walls,  if  there  be  no  tie,  is  H=  |  Q  cotg.  $,  when  6  is 
the  inclination  of  the  rafters  or  braces  JIB  and  JilBl  to  the  horizon. 
As  this  angle  is  frequently  a  small  one,  the  thrust  is  considerable, 
and,  therefore,  care  must  be  taken  with  the  foot  fastenings  (see 
Vol.  II.  §  17).  The  scantlings  of  the  braces  and  collar  beams  must 
be  fixed  by  the  rules  in  Vol.  I.  §  206,  &c.,  so  that  they  shall  resist 
flexure  and  fracture,  when  exposed  to  forces 


S  = 


2  sin.  «' 


and  H  =  J  Q  cotff. 


The  force  Q  depends  on  the  loading  of  the  bridge  or  roof.     If  the 


48 


TIMBER  BRIDGES. 


load  be  uniformly  diffused,  we  shall  do  best  to  assume  that  each  post 
carries  ^,  and  each  side  wall  ^  of  the  load. 

Example.  Suppose  the  trussed  bridge  in  Fig.  38,  designed  as  one  of  two  for  a  60  feet 
span  and  12  feet  wide  bridge:  suppose  each  square  foot  of  the  bridge  together  with  its 
load  weighs  50  Ibs.,  the  weight  of  the  bridge  is  12  X  60  X  50=  36000  lbs.,and  the  load 


on  the  queen-posts : 


36000 


Fie.  39. 


=  12000.     Therefore,  for  an  inclination  of  the  rafters  of 

22$°,  the  horizontal  pressure  =  $  12000 
cotg.  22J°  =  6000  X  2,414-2  =  14485 
Ibs.,  and  the  thrust  through  each  rafter 


of  these  strains  come  on  the  pieces  of 
each  of  the  two  trusses,  so  that  on  each 
collar  there  would  be  7242.5  Ibs.,  and 
on  each  rafter  7839,5  Ibs.  If  we  take 
ihe  resistance  of  wood  (Vol.  I.  §  206) 
at  7400  Ibs.,  and  if  we  strain  only  to 
Jjr  of  the  absolute  strength,  we  get 
for  the  section  of  each  collar  beam 
„  7242,5  .  20  1448,5 


square  inches,  and  for  each  brace  or  rafter 


7839,5  .  20 


7400 

15679 

74 


74 


19,6 


21,2  square  inches. 


Fig.  40. 


Remark.  More  composite  trusses,  as 
indicated  in  Figs.  39  and  40,  are  calcu- 
lated in  the  same  manner  as  the  above. 
In  each  of  these  it  may  be  assumed 
that  each  of  the  four  posts  or  uprights 
carries  one-fifth  of  the  entire  load,  and 
that  the  remaining  fifth  rests  immedi- 
ately on  the  side  walls.  In  the  con- 
struction shown  in  Fig.  40,  the  direc- 
tions of  the  different  rafters  are  not 
optional,  but  dependent  one  upon  the 
other.  If  Q  be  the  weight  on  each  post, 
and  >  the  inclination  of  the  brace  J5C, 
and  J,  that  of  JIB,  the  horizontal  thrust 

H  =  Q  cotg.  »  =  (Q  -f  Q)  rot^.  *n  hence  mt8-  *  =  2  cotg.  *, 
or  tang.  J,  =  2  tang.  J 

§  26.   Timber  Bridges.  —  The  framings  in  the  foregoing  section 
support  the  road-way  or  ceiling  by 
Fig.  41.  suspension,  but  there   are   trusses 

applied  for  bridges,  which  support 
the  road-way  on  the  opposite  prin- 
ciple of  sustaining  them.  In  these 
latter,  the  distribution  of  the  pres- 
sure takes  place  exactly  as  in  the 
former.  In  the  simple  case  shown 
in  Fig.  41,  we  have  from  the  verti- 
cal force  Q  acting  at  the  centre  of  the  bridge  AAV  the  horizon- 
tal thrust  H  =  |  Q  cotg.  8,  and  the  strain  on  the  spear  or  strut 

BC  =  S  =  i     ^ 

"  sin.  o 
example,  Fig.  42,  the  forces  are  the  same,  but  in  this  case  Q  may 


,  when  6  is  the  inclination  of  the  strut.     In  the 


TIMBER  BRIDGES. 


49 


be  taken  at  ^  of  the  whole  load,  whilst  in  the  case  Fig.  41,  Q  =  J 
the  load.     The  piece  CC^  in  Fig.  42  is  termed  a  straining  till.     If 

Fig.  42. 


Fig.  43. 


there  be  a  double  set  of  struts  or  spears,  as  indicated  in  Fig.  43, 
there  are  four  struts,  and  it  may  be  assumed  that  each  carries  one- 
fifth  of  the  whole  load,  or 
Q  =  I  G.  To  prevent  de- 
flection of  long  spears,  braces 
or  counter-braces  JIT),  ^1D1 
are  added,  particularly  when 
there  are  several  sets  of 
spears.  The  distribution  of 
the  pressure  in  the  case  of 
spears  of  unequal  length  be- 
ing used  as  in  Fig.  44,  is  to 
be  taken  as  exactly  the  same 
as  in  Fig.  43,  only  that  in 
these  the  braces  or  suspend- 
ing posts  CD,  C1Dl  become 
the  more  requisite  as  the 
struts  come  to  have  consi- 
derable length.  It  is  proper 
to  take  the  weight  of  all  the 
parts  into  calculation,  and  to 
reckon  that  half  the  weight 
of  each  part  acts  at  its  end. 
The  centerings  for  bridges 
afford  the  most  frequent  ap- 
plication of  the  kind  of  fram- 
ing we  are  now  considering. 
Figs.  45  and  46  represent 
two  such  centres.  The  pressure  which  each  simple  frame  j3BB1.fll 
or  JlBJll  undergoes  and  has  to  resist,  may  easily  be  determined 
by  calculating  the  weight  of  the  part  of  the  arch  bearing  upon  it. 

VOL.  II. — 5 


50 


POSTS. 


Fie.  4fi. 


strain  along  the  spears  S 


Q  cos. 


If  the  two  spears  abutting 
on  each  other  have  different 
inclinations  to  the  horizon, 
as  in  the  construction  shown 
in  Fig.  46,  the  strain  on  them 
is  of  course  unequal.  If  the 
angles  of  inclination  of  two 
such  spears  =  8  and  81?  and 
if  the  vertical  pressure  at 
the  abutting  joint  =  Q,  the 
6  cos.  S 


the  horizontal  thrust  of  both  =  H 


-  Sj  sin.  (8 

Q  cos.  5  cos.  $. 


and 


Fijj.  47. 


sin.  («  +  5J 

§  27.  Hoofs.  —  In  roofs,  collar 
beams  are  applied  to  prevent  deflex- 
ion of  the  rafters,  as  also  queen-posts, 
braces,  &c.,  and  the  nature  of  the 
forces  may  be  traced,  as  in  Figs.  47, 
48,49. 

§  28.  Posts.— The  strength  of  pil- 
lars and  posts  subjected  to  tensile  or 
compressive  strains,  when  these  act 
in  the  direction  of  the  axis,  have  been  investigated,  Vol.  I.  §  183  to 
§  206.  It,  however,  not  unfrequently  happens,  that  the  forces  act 

Fig.  48.  Fit  40. 


out  of  the  axial  direction,  and  we  shall,  therefore, 
examine  this  case.  EF  in  Fig.  50,  represents  a 
suspending  post  to  which  a  tensile  strain  P  is 
applied  excentrically.  Let  F =  the  area  Z,  the 
length  EF  of  the  post,  a  =  the  leverage  FH,  or 
the  distance  of  the  direction  of  the  force  from  that 
of  the  axis.  Prolong  FH  in  the  opposite  direc- 
tion, and  make  FL  =  FH  =  a,  and  conceive  that 
in  L  two  equal  and  opposite  forces  J  P,  —  J  P 
act :  there  results  an  axial  force  FP  =  P,  and 
a  couple  J  P, —  J  P.  The  former  extends  all  the 


POSTS.  51 

p 

fibres  uniformly  by  a  quantity  xx  =  —  —  -  .  I,  but  the  latter  extends 

.r  .  hi 

the  fibres  unequally  on  one  side,  and  compresses  them  unequally  on 
the  other.  If  the  post  be  rectangular,  with  the  sectional  dimen- 
sions b  and  A,  where  h  is  in  the  same  plane  as  a,  the  moment  of  the 
force  : 

E  (Vol.  I.  §  191), 
but  the  extension  or  compression  of  the  fibres  at  the  distance  1  from 

the  axis  :  \  =  -  ,  and  that  of  the  extreme  fibres 
o  h  E 

=  -  .  x2  =  —  —  ,  therefore  the  greatest  extension  : 
2  oh2  E 


But  for  the  force  K  producing  rupture  :  _  =  -,  hence  the  modulus  of 

E       I 
strength  : 

K  =  —  (  1  +  _?),  and  inversely  : 
oh  \  h  / 

P  »     bk     .  K. 


If  the  post  be  cylindrical,  and  its  radius  =  r,  we  have  (Vol.  I. 
195). 


,  hence  „ 
and  the  longitudinal  extension  : 


If  the  force  act  at  the  periphery  of  the  post,  we  have  in  the  first 
case  a  =  J  h,  and  in  the  second  a  =  r,  and,  therefore,  for  the  rectan- 
gular section  P  =  —  K,  and  for  the  cylindrical  P  =  1tr2R ,  Thus, 

theoretically,  a  rectangular  post  will  carry  only  £,  and  a  cylindrical 
one  only  £  when  loaded  in  the  direction  of  the  side  of  what  it  will 
carry  when  fairly  loaded.  Experiments  on  cast  iron  give  results  of 
J  instead  of  J  for  rectangular  columns. 

The  same  laws  apply  to  the  uprights  J1C,  Fig.  51,  but  then  x  must 
be  taken  as  the  greatest  compression. 

If  the  column  be  inclined,  as  in  Fig.  52,  and  if  its  foot  make  an 
angle  a,  with  the  horizon,  we  may  decompose  P  into  two  others 
Pl=  P  sin.  a,  and  P2  =  P  cos.  a,  and  in  the  equations  for  xt  and  x2, 


52 


POSTS. 


we  must  substitute  P  sin.  a,  for  P,  and  besides  this  the  extension  x3 

produced  by  the  normal  force 
P  cos.  a,  has  to  be  introduced. 
If  we  substitute  P  cos.  a  for 

P,  and  I  for  a,  we  obtain  — -2 

for  the  greatest  extension  or 
compression  produced  by  the 
force  P  cos.  a,  and  hence  for 
rectangular  sectioned  beams 
this  extension  or  compression. 


—    n       _I_  2       i^  3    _  I     /  I     jJ/' 

1  "*"  ~2~  '    T~ 5=  ^  IVTl  "Tl2/ 


and  therefore  the  tension 


If  the  arm  Fff  be  on  the  up-side  of  the  beam,  as 
shown  in  Fig.  53,  we  then  have: 

p  = bhK 


and  for  round  columns  the  expression  becomes: 


A       4a\     .  4/ 

(  1  H J  Sin.  a  4 COS.  a. 

\    —  r  J  r 

§  29.  If  a  loaded  beam  AB,  Fig.  54,  rests  upon  two  uprights,  the 

load  P  bears  upon  each  in  the  proportion  J  P  on  J1D,  and  -1  P  on 
BE,  when  71}  /a,  and  Z,  represent  the  lengths  .##,  C/2,  and  C5  re- 

Fiz  54.  Fis  S5.  Fig.  56. 


spectively.  If  a  similar  beam  rests  upon  three  or  more  uprights, 
the  pressure  on  each  can  only  be  determined  by  aid  of  the  theory  of 
the  elastic  resistance  of  materials.  If  weights  P  and  P  act  at  the 


POSTS.  53 

centre  of  the  lengths  AC  and  BC,  and  if  we  assume  that  the  one 
part  AC  is  independent  of  the  other  part  J3C,  the  pressure  on  the 
centre  upright  =  P,  and  that  on  each  of  the  others  =  £  P.  But  if 
we  consider  the  beam  as  an  entire  piece,  the  circumstances  are  dif- 
ferent. 

When  a  beam  fastened  by  one  end  into  a  wall  AB,  Fig.  56,  sup- 
ports a  weight  P  at  (7,  and  is  supported  at  the  other  end  -B,  the 
beam  forms  an  elastic  curve,  horizontal  at  A,  but  inclining  upwards 
at  B.  For  simplicity's  sake,  let  us  assume  P  as  acting  at  the  middle 
C,  and  put  the  length  AB  =21  The  deflexion  BT  of  the  outer 
half  CB  is  equal  to  the  deflexion  of  the  inner  half  AD  =  BE  plus 
the  tangent  distance  TE.  But  according  to  Vol.  I.  §  189,  the  height 

P  I3 

BT  =  —  i  —  ,  if  Px  be  the  force  on  the  end  B  required.    Again  the 
3  WE 

deflexion: 

AD=  Z^_^L_  (V  P-  V  I3}-  -**--**iL, 
3  WE      iTTCE  ~~  3  WE        6  WE' 

and  the  tangent  distance: 


and  hence  it  follows: 

if  =  T~^  +  T-^'  orl6Pl  =  5  P'  therefore  Pl  =  l5g  P' 

According  to  this  view  of  the  matter,  the  support  B  bears  T5g  P, 
and  the  point  of  fixture  A  ig  P.  The  same  relations  obtain  in  the 
case  of  a  beam  supported  by  three  uprights,  when  the  ends  A  and  B 
are  free  to  move  up,  but  the  middle  part  C  kept  horizontal.  The 
uprights  under  A  and  B  carry,  therefore,  each  T5g  of  the  weight  P, 
whilst  the  centre  post  carries  f  §  P. 

If  the  supports  be  inclined  as  shown  in  Fig.  57,  there  arises  a 
horizontal  thrust  H  =  J  P 

cotg.  8,  with  which  the  feet  Fig.  57.  Fig.  58. 

of  the  posts  tend  to  spread. 
If,  again,  a  beam  resting 
upon  two  uprights  be 
strengthened  by  two  braces 
as  shown  in  Fig.  58,  we 
may,  though  only  as  an  ap- 
proximation, assume  that  at 
each  end  A,  Ar  a  pressure 
—  352  P  acts;  whilst  on  each 
point  C,  Cv  there  is  a  pressure  of  £|  P.  If  «  be  the  angle  of  inclina- 
tion BCA  of  the  braces  BC,  the  horizontal  thrust  in  C  and  -B,  =  i  £  P 

cotg.  8,  and  the  thrust  along  the  brace  l|  --     If,  again,  I  be  the 

sin.  « 
whole  length  AD,  and  ?a  the  part  5Z)  of  the  support  measured  up  to 

the  brace,  the  horizontal  strain  on  the  upright  =  --  .    ,8f  P  cotg.  i, 

I 

5* 


54 


BRACES  OR  bTRUTS. 


and,  therefore,  the  column  has  not  only  to  bear  the  vertical  pressure 

n  7 

— ,  but,  likewise,  a  horizontal  force  =  J  T8f  .  P  cotg.  8,  creating  flex- 
ure round  B.  In  order,  therefore,  to  insure  the  sufficiency  of  such 
a  frame,  the  formula : 

i  P  =  bh K  :  ( 1  +  6  .  I f  ^"^P*1  cotff.t}, 

must  be  satisfied. 

§  30.  Braces  or  Struts. — Fig.  59  shows  a  case  of  frequent  occur- 
rence.    Where  a  beam  JIB,  fixed  in  a  wall  or 
59.  otherwise  at  one  end,  loaded  at  the  other,  is 

strengthened  by  a  brace  or  strut  CD.  Let 
JIB  the  length  of  the  beam  =  /,  and  the  part 
AC  =  lv  the  inclination  of  the  beam  =  a,  and 
that  of  the  strut  =  8.  From  the  load  P  there 
arises  a  vertical  pressure  in  C  downwards : 

V  =  -  P,  and  a  vertical  pressure  at  A  upivards : 
P-     The  first  vertical  pressure 


downwards  resolves  itself  into  two  forces  along  the  axes  of  the 
pieces : 

«,  V  cos.  8  I P  cos.  8  , 

S  =  — : — . -  =  - — : — r,  and 


sin.  (8  —  a) 

V  COS.  a 


ll  sin.  (8  —  o) 
I  P  cos.  o 


sin.  (8  —  a)        Zj  sin.  (8  —  a) 

The  case  shown  in  Fig.  60,  where  the  beam  is  supported  by  a  tie- 
brace,  is  to  be  treated  in  a  manner  exactly  similar  to  the  above.  In 
most  cases,  the  beam  JIB  is  horizontal,  or  o  =  0°,  then  we  have : 


S=  V  cotg.  8 


IP 


cotg.  8  and  S1 


IP 


Fiz.  60. 


Fig.  61. 


sin.  8 

The  dimensions  of  the 
brace  have  to  be  determined 
in  proportion  to  the  strain 
tfj  acting  on  it,  and  that  of 
the  beam  with  reference  to 
strain  S  compressing  it, 
and  likewise  the  cross  strain 
arising  from  P,  acting  with 
the  moment  P  (I — ?.).  Hence 
(§28): 


and  by  this  equation,  the  section  b  h  of  the  beam  must  be  deter- 
mined. For  the  case  shown  in  Fig.  61,  we  have  to  find  the  strain 
on  the  upright.  The  part  DE  of  the  upright  is  compressed  by  the 


COMPOUND  BEAMS. 


55 


force  P,  and  strained  across  by  the  moment  PL  therefore  we  must 

7      7       Tf 

put  P  =  -  —  ,  in  order  to  get  the  required  section  b  h.    The  piece 


J1D,  on  the  other  hand,  is  under  a  tensile  strain 


P,  whilst 

the  cross  strain  is  the  same,  as  for  lower  part  ;  we  have,  therefore,  in 
this  case  :  b  h  K 


If  at  the  foot  of  the  upright  there  be  placed  a  strut  FG,  this 

I  P 

would  take  up  the  strain  S  =  -  ,  if  a  be  its  inclination,  and 
a  cos.  a 

PI 

a  =  EF,    and   the   force   S1  =  —  tang,  a  passes  through  the  up- 

h 

rights.  Hence  the  part  EF  of  the  upright  is  strained  by  a  force 
=  P  —  St  or  tfj  —  P,  the  former  when  a  cotang.  a  <  Z,  and  the 
latter  when  a  cotang.  o  >  Z,  or  according  as  the  strut  falls  within, 
or  beyond,  the  point  of  suspension. 

Example.  In  the  framing,  Fig.  61,  suppose  P  =  1500  Ibs.,  jiB  =  12  feet,  the  upright 
EJ1  =  24  feet,  the  inclination  of  the  braces  =  45°,  and  the  horizontal  projection  of  each 
=  6  feet  ;  required  the  necessary  strength  for  the  frame. 

The  braces  have  strains  : 

c,  IP  12  .  1500          3000          ._  ._  ., 

S,  =  -  =  -  =  -  =  4243  Ibs.  to  withstand. 

liSin.X       6  sin.  45°         0,7071 
Taking  7400  as  modulus  of  strength,  we  get,  allowing  20  times  absolute  strength,  the 


section  of  each  brace 


4243 

-  .  20  =  11.5  square  inches.     For  the  beam  we  may  take 

according  to  Vol.  I.  §  198,  K  =  12000  Ibs.,  for  breaking  across  is  here  most  likely  to 
occur.     Allowing  20  times  the  absolute  strength,  we  have  to  put: 


If  now  we  make  the  depth  of  the  beam  double  its  breadth,  we  get  : 

2  62  =  5   (l  _|_  -^  ,  or  63  —  f  b  =  */.     From  this  we  get  the  breadth  of  the  beam 

3,1  inches,  and  the  depth  6,2  inches.     For  the  upright,  that  is,  for  the  centre  part,  by 
similar  reasoning  we  get  : 

20  .  1500  =     r2000bh    ,  that  is  JS^.  =  |.  or  6  h  =  f  +  1^, 
612  !•>  h 


and  if  in  this  case  we  make  h  ^  2  6,  we  get  63  — 
h  =  7,4  inches. 


b  =  45,  from  which  b  =  3,7,  and 


§  31.  Compound  Seams.  —  Beams  laid  upon  one  another,  and 
united  only  by  bolts,  Fig.  62,  have  a  resistance  equal  only  to  the 
sum  of  the  resistances  of  the  individual  beams.  If  the  beams  only 
abut  on  each  other,  as  in  Fig.  63,  and  the  butting  joints  be  made  to 


Fig.  62. 


FIK.  63. 


56 


COMPOUND  BEAMS. 


break  joint,  the  strength  of  one  beam  is  lost  to  the  whole.  If  the 
beams  be  morticed,  and  tenoned  as  in  Figs.  64  and  65,  well  strapped 
together,  the  strength  of  the  combination  is  almost  equal  to  that  of 
a  solid  beam  of  the  same  dimensions. 


Fig.  64. 


Fig.  65. 


Beams  are  frequently  built  in  this  manner,  to  get  great  strength. 
The  resistance  of  the  elements  of  a  beam  increase,  as  their  distance 
from  the  neutral  axis.     If,  therefore,  we 
66  separate   two   beams  by  thick  tenons  or 

wedges,  and  then  strap  or  bolt  them  to- 
gether, as  in  Fig.  66,  their  strength  is  con- 
siderably increased.     If  b  be  the  breadth 
and  h  the  depth,  I  the  length  and  a  the  distance  between  the  two 
beams,  the  strength  of  the  combinations  (Vol.  I.  §  200)  is: 
P  -  /(a+2h}3—a\   bK. 
~\    l(a+2h) 


If,  for  example,   a  =   2  h,  then  P  =  14 

772 


773         V 

-  .  —  ,  whereas  P 
I       6 


=  4  -  .  —  ,  if  the  two  beams  had  only  been  morticed  together.* 

I         6 

The  same  relations  obtain  in  the  beam,  shown  in  Fig.  67,  united 
by  St.  Andrew's  crosses  or  lattice-fram- 
-  67-  ing.    In  like  manner,  we  determine  the 

strength  of  wooden  beams,  composed 
of  curved  pieces,  as  in  the  bridge,  Fig. 
68,  but  it  must  be  strictly  borne  in 
mind,  that  wooden  framings  lose  much 
of  their  strength  by  deflexion.  A  principal  advantage  of  such  con- 
structions is,  that  they  are  more  stiff,  and  less  liable  to  vibrate  than 

Fig.  68. 


*  Obvious  as  is  the  truth  of  this  statement,  and  easy  as  is  its  application  in  practice,  it 
is  singular  that  so  little  use  is  made  of  it  in  the  construction  of  timber  bridges  and  other 
building*  in  this  country.  It  is  evidently  applicable  to  the  double  beam  arches,  often 
inserted  in  the  socalled  arch  and  truss  bri.'ges. — AM.  ED. 


COMPLEX  STRUJTURES. 


57 


simple  beams;  and  that,  as  they  act  only  vertically  on  their  points 
of  support,  they  require  no  abutments,  properly  so  called. 

Curved  beams,  as  shown  in  Fig.  69,  have  been  frequently  applied 


Fis  60. 


in  cast  iron  structures,  and  cast  iron  arches,  as  in  Fig.  70,  are  a  very 
usually  employed  bridge  material.     To  judge  of  the  strength  of  such 

Fig.  70. 


a  structure,  its  line  of  resistance  must  be  determined.  If  this  fall 
everywhere  within  the  arch,  it  shows  that  there  is  no  cross-strain  on 
the  material,  but  only  compression ;  but  if  the  line  of  resistance  fall 
ivithout  the  arch,  the  weak  point  is  where  it  runs  furthest  from  the 
arch,  and  the  resistance  of  the  material  to  cross-strains,  is  that  upon 
which  the  stability  of  the  structure  depends. 

[Proof  of  the  strength  of  Complex  Structures  by  means  of  Models. — 
The  plan  of  solving  questions  in  practical  mechanics  and  engineer- 
ing by  faithfully  constructed  models,  presents  the  very  obvious  ad- 
vantage of  substituting  the  moderate  cost  of  experiment  for  the  often 
burdensome,  sometimes  ruinous,  expense  of  experience.  The  condi- 
tions to  be  fulfilled  in  constructing  models,  so  as  to  give  reliable 
information  in  regard  to  the  action  or  the  stability  of  structures, 
may  be  stated  as  follows: — 

1st.  An  entire  correspondence  must  exist  in  the  model,  (at  least, 
of  all  essential  parts,)  to  the  scale  of  dimensions  and  weights  on  which 
it  is  proposed  to  represent  the  structure. 


58  COMPLEX  STRUCTURES. 

2d.  Identity  not  only  in  the  nature,  but  also  in  the  condition  of 
materials  employed  in  the  model  and  structure  respectively. 

3d.  Proportional  accuracy  in  forming  junctures ;  and  proportional 
tension  given  by  tightening  screws,  keys,  wedges,  and  other  me- 
chanical means,  by  which  the  parts  are  compacted  together. 

In  testing  the  model,  modes  of  introducing,  distributing,  and  with- 
drawing loads,  conformable  to  those  which  practice  will  involve  in 
regard  to  the  structure,  must  be  observed,  so  as  to  subject  the  model 
to  shocks,  jars,  inequality  of  pressure  and  irregularities  of  applica- 
tion, at  least  proportional  to  those  which  the  structure  will  be 
required  to  sustain. 

Supposing  the  model  of  a  bridge  to  have  been  constructed  accord- 
ing to  the  above  requirements,  it  might  be  used  for  either  of  the  two 
following  purposes: — 

1.  To  determine  what  weight  the  structure  will  bear  when  under- 
going a  given  deflexion,  or  when  on  the  point  of  breaking. 

2.  To  ascertain  whether  the  principle  of  construction  be  adequate 
to  furnish  a  bridge  of  the  proposed  dimensions,  and  materials  that 
can  fulfil  the  specified  duty. 

As  a  beam  or  bridge  of  uniform  dimensions  throughout  will  bear 
half  as  much  weight  accumulated  at  the  centre  as  it  could  sustain  if 
distributed  throughout  its  length,  the  simplest  mode  of  arriving  at 
the  result  desired  is  to  determine  and  apply  to  the  centre  of  the 
model  a  weight  which  shall  represent  one-half  the  load  supposed  to 
come  upon  the  structure. 

The  following  formula  applies  to  the  loading  of  the  model  at  its 
centre. 

Let  b  =  the  length,  in  feet,  of  the  model  between  the  points  of 
support;  p  the  weight  in  pounds  which  the  model  is  to  sustain  at 
the  centre,  representing  a  load  uniformly  distributed. over  its  length; 
w  =  the  weight  of  so  much  of  the  model  as  lies  over  the  clear  open- 
ing between  its  piers ;  r  =  the  ratio  of  dimensions  between  the 
structure  and  the  model;  P  =  the  load  which  the  structure  must  be 
able  to  bear,  when  accumulated  at  the  centre.  Then  it  is  evident 
that  rl  =  the  length  of  structure  between  the  piers.  Since  the 
relative  resisting  powers  of  similar  beams  or  bridges  are  as  the 
second  powers  of  their  corresponding  dimensions,  .'.  r2  :  1  :  :  resist- 
ing power  of  the  structure  :  resisting  power  of  the  model.  Hence, 
r2  (p  +  \  w)  =  the  absolute  resisting  power  of  the  structure.  Also, 
since  the  weights  of  similar  structures  are  as  the  third  powers  of 
their  corresponding  dimensions — or,  what  is  the  same  thing,  as  the 
third  powers  of  their  ratios  of  dimensions — therefore  r3  w  =  the 
absolute  weight  of  the  structure;  so  that  the  weight  P,  which,  by 
supposition,  the  structure  can  bear,  accumulated  at  its  centre,  will 
be  its  absolute  resisting  power,  diminished  by  half  its  own  weight. 

Hence,  P  =  *  (p  +  |)  _  ^  .  *f  _  H+Hl  =  ,.„_ 


CHAIN  OR  SUSPENSION  BRIDGES. 


59 


But  as,  by  supposition,  P  is  known,  and  it  is  desired  to  find  p,  the 
conversion  of  the  last  formula  gives  p  =  —  -\ (r  —  1)  [2]. 

Example.  It  is  required  to  construct,  on  a  given,  plan,  a  bridge  having  a  clear  opening 


between  the  piers  of  150  feet,  and  capable  of  sustaining  two  tons  per  foot  of 
or  300  tons  in  all,  equally  distributed  over  its  surface.  A  model  is  made  on  tl 
one  inch  to  the  foot,  and  weighing  136.3  pounds,  exclusive  of  the  part  w 
directly  upon  the  abutments.  It  is  required  to  find  what  number  of  pound 
suspended  from  the  centre  of  the  model,  in  order  to  prove  whether  any  b 


ts  length, 
e  scale  of 
lich  rests 
must  be 
idge  con- 


structed on  the  plan,  with  the  relative  dimensions  and  of  the  materials  used  in  the  model, 
will  bear  the  load  above  specified. 

Substituting  the  values  of  the  several  symbols  in  the  second  of  the  above  equations, 
viz:  ^  =  ^+^(r-l),  we  obtain  jp  =  3°°X  2240  136_3Q82 

r3  T  2   V  12  X  12  2 

pounds;  and  twice  this  number,  or  6164  pounds,  is  the  weight  which  the  model  ought 
to  bear,  when  distributed  uniformly  over  its  surface.] 

§  32.  Chain  or  /Suspension  Bridges. — Suspension  bridges  involve 
considerations  distinct  from  the  principle  of  the  stability  of  either 
stone,  wood,  or  cast  iron  bridges,  inasmuch  as  the  road-way  is  sus- 
pended from  chains  or  ropes,  or  is  supported  upon  these.  The 
former  is  the  more  frequent  construction.  Chains  or  cables  drawn 
up  with  considerable  force,  between  two  or  more  piers  or  supports, 
pass  over  these  to  fastenings  in  rock  or  masonry,  as  shown  in  Fig. 
71.  The  chains  are  formed  of  malleable  iron  bars,  united  by  pins 

Fig.  71. 


or  bolts :  and  cables  of  iron  or  steel  wire,  laid  parallel  or  twisted 
together,    are   frequently   employed   instead   of   bar-chains.     The 

Fig.  72. 


60  CHAIJs  OR  SUSPENSION  BRIDGES. 

dimensions  of  the  links  or  bars,  depend  upon  those  of  the  bridge. 
In  large  bridges  they  are  made  about  1  inch  thick,  from  3  to  9 
inches  deep,  and  from  10  to  16  feet  long.  Usually,  several  sets  of 
bars  are  hung  together,  forming  a  compound  chain  united  by  coupling 
plates  and  bolts,  as  shown  in  Fig.  72  (or  without  coupling  plates, 
according  to  Mr.  Howard's  patent  plan).  Wire  cables  are  com- 
posed of  wires  of  from  ^  to  J  of  an  inch  in  diameter,  and  are  made 
of  any  requisite  diameter,  varying  from  £  an  inch  to  3  inches.  The 

suspending  rods  consist  of  wrought 

Fig-  74.  Fig.  75.          iron  rods,  or  of  wire  ropes.     The 

rods  JIB,  ^Bj,  are  hung  by  pins 
passing  through  the  coupling  plates 
as  shown  in  Fig.  73,  and  suspend- 
ing ropes  are  attached  as  shown 
in  Figs.  74  and  75,  by  means  of 
shackles  with  eyes,  or  by  a  simple 
loop.  The  cross-beams  of  the  road- 
way C,  Cj  are  sometimes  fastened 
to  the  suspending  rods  as  shown  in 
Fig.  74,  sometimes  as  shown  in  Fig. 

73.  The  rod  goes  either  through  the  beam,  and  is  then  fastened 
by  a  nut  resting  on  a  metal  plate,  or  washer,  or  a  stirrup,  or  strap 
is  put  over  the  beam,  a  hook  on  the  upper  side  of  which  goes  into 
the  eye  of  the  shackle  of  the  suspension  rope,  or  into  the  loop  formed 
on  it.  Upon  the  cross-bearers  longitudinal  beams  are  laid,  and  these 
are  covered  with  three  inch  planking,  and  again  three  inch  cross 
planking,  according  to  circumstances,  and  upon  this  road-metalling, 
&c.,  is  laid.  In  general  there  are  two  systems  of  chains,  one  above 
the  other,  on  each  side  of  the  bridge,  and  hence  the  number  of  sus- 
pension rods  is  twice  the  number  of  joints  in  any  one  chain  system. 
The  distance  from  centre  to  centre  of  suspending  rods  is  about  five 
feet. 

The  parapet  of  the  bridge  ought  to  be  framed  so  as  to  give  the 
greatest  stiffness  to  the  road-way.* 

The  width  of  road-way  depends  on  the  purposes  which  the  bridge 
is  to  subserve.  There  should  be  3  feet  at  least  for  a  foot-path,  and 
7  to  7 J  for  a  carriage  way.  For  a  bridge  for  ordinary  traffic,  a  total 
width  of  25  feet  between  the  parapets  is  sufficient. 

§  33.  The  versed  sine  of  the  arc  of  suspension  bridges,  is  generally 
small  in  proportion  to  the  cord,  varying  from  4  to  ^  and,  there- 
fore, the  strain  on  the  chain  is  very  great  (Vol.  I.  §  144).  The  piers 
on  which  the  chains  pass,  and  the  fastenings  by  which  chains  are 
held  must  withstand  very  considerable  forces,  and  hence  piers  of 
great  stability,  and  abutments,  or  rather  anchorage,  of  great  resist- 
ance must  be  provided.  The  span  of  suspension  bridges  is  regu- 
lated by  various  circumstances.  A  series  of  smaller  spans  is  often 
much  more  economical  than  one  or  more  large  spans  to  cover  the 
same  interval. 

*  See  Appendix. 


CHAIN  OR  SUSPENSION  BRIDGES. 


61 


The  Menai  bridge  in  England,  the  two  bridges  at  Fribourg  in 
Switzerland,  the  bridge  at  Roche  Bernard  in  France,  the  bridge  over 
the  Danube  at  Ofen,  are  examples  of  large  spans  of  from  600  to  720 
feet ;  whilst  there  are  innumerable  instances  of  less  span  in  every 
country.  If  the  chain  be  not  equally  strained  on  the  two  sides  of 
the  pier,  which  always  occurs  when  one  side  only  is  loaded,  the  chain 
slides  forward  towards  the  side  on  which  there  is  the  greater  load. 
As,  however,  there  would  arise  considerable  friction  between  the  rope 
and  the  head  of  the  pier,  under  the  pressure  of  the  resultant  force 
being  on  it,  the  pier  must  have  stability  to  counteract  a  force  equal 
to  this  friction.  To  prevent  this  action,  special  contrivances  are 
adopted  for  diminishing  the  friction.  These  means  consist,  either  in 
passing  the  chains  over  rollers  or  pullies,  Fig.  76,  which  reduces  the 
sliding  friction  to  a  rolling  friction  on  a  small  axle,  or  the  chains 
pass  over  a  sector  which  rocks  on  the  head  of  the  pier,  inclining  to 
one  side  or  the  other  as  external  forces  act  upon  it ;  or,  lastly,  the 
pier  is  made  as  a  column  rocking  on  its  foot,  or  on  a  horizontal  axis 
at  its  foot.  That  the  resultant  of  the  forces  acting  on  the  chain  may 
press  vertically  on  the  pier  head,  and  thus  be  least  strained  by  it,  it 
is  necessary  that  the  parts  of  the  chain  on  each  side  of  the  pillar 
should  have  equal  inclinations  to  the  horizon.  If  this  equality  can- 
not be  obtained,  as  is  not  unfrequently  the  case  for  the  land  piers 
of  bridges,  the  piers  must  be  considerably  strengthened. 


Fig.  76. 


Fig.  77. 


To  fasten  the  ends  of  the  chains  to  the  land,  various  devices  have 
been  practised,  the  general  plan  of  which  is  to  carry  the  chains  by 
wells  or  drifts  into  the  rock  or  soil,  and  there  to  fasten  them  _  to 
broad  iron  or  wooden  piles,  or  planking  as  at  JlB,  Fig.  77,  which 
abut  upon  substantial  retaining  walls  of  masonry,  or  against  an  arch, 
or  against  the  rock  itself.  The  fastenings  can  thus  be  examined  at 
any  time,  and  adjusting  wedges  for  compensating  the  influences  of 
expansion  and  contraction  be  conveniently  manipulated. 

Remark.  On  the  subject  of  suspension  bridges,  the  most  complete  treatise  is  that  of 
Navier,  "Rapport  et  Memoire  sur  les  ponts  suspendus,  Paris,  1823."    The  papers  of  Mr. 
VOL.  II. — 6 


CHAIN  OR  SUSPENSION  BRIDGES. 


Davies  Gilbert,  in  the  "Transactions  of  the  Royal  Society  of  London,  1826,"  are  import- 
ant in  the  history  of  these  bridges.  In  Moseley's  "  Engineering  and  Architecture"  there 
is  a  very  elegant  investigation  of  the  properties  of  these  structures.  The  treatise  of 
Drewry  on  "Suspension  Bridges,  1832,"  is  a  very  excellent  resume  of  the  general  prac- 
tice in  respect  to  suspension  bridges.  The  account  of  the  suspension  bridge  over  the  Vi- 
laine,  at  La  Roche  Bernard,  by  Leblanc,  Paris,  1841,  is  very  instructive.  There  is  a 
treatise  of  Seguin,  "  Memoire  sur  les  ponts  en  fil  de  fer,"  worthy  of  attention.  There 
are  many  memoirs  in  the  ''Annales  des  ponts  et  chaussees"  on  this  subject;  and,  in  the 
volume  for  1842,  there  is  an  account  of  a  bridge  made  of  ribbons  of  hoop  iron. 

§  34.  The  curve  formed  by  the  chain  or  cable  of  a  suspension 
bridge,  lies  between  the  parabola  and  the  catenary,  and  is  very 
nearly  an  ellipse.  The  parabola  approximates  the  curve  in  the 
loaded  bridge,  the  catenary  in  the  unloaded  (compare  Vol.  I.  §  144 
and  §  145,  &c.).  We  shall  consider  the  curve  as  a  parabola,  or  the 
bridge  in  its  loaded  state. 

If  the  two  points  of  suspension  B  and  J),  Fig.  78,  of  a  chain,  be 

Fig.  78. 


on  the  same  level,  and  if  BD  =  25,  and  A  C  the  versed  sine  or 
height  of  the  arc  =  a,  and  the  angle  CBT =  CDT=  a,  then 

^.a=£I=^(Vol.I.  §144). 
z>  C          0 

If  the  points  of  suspension  be  at  different  levels,  as  in  Fig.  79,  the 

Fig.  79. 


apex  of  the  curve  is  not  in  the  centre,  and  the  ends  of  the  chain 
have  different  inclinations.  If  we  put  the  co-ordinates  AC  and  BC 
=  a  and  5,  and  the  co-ordinates  AF  and  FD  =  al  and  bv  we  put 
the  whole  span  BE  =  s,  and  the  difference  of  DE  =  h,  we  have : 

7i  =  a  —  a,,  s  =  b  +  5,,  and  —  =  _ ,  we  have,  therefore,  from  7i, 

«i        *i2 
s,  and  a: 


DIMENSIONS  OF  THE  CHAINS  AND  ROPES.  63 


1  ,  a,  =  a  —  h,  2  ,  b  =  -  *_^, 


J-  !+  !- 

\|  a  \ja, 


and  for  the  angles  of  inclination  a  and  ax: 

2  a          T  ..  2  a, 

r.  a  =  -  ,  and  tang.  ^  =  —  -1. 


, 

The  length  of  the  parts  of  the  chain  AB  =  I  and  J1D  =  lv  is 
expressed  by: 

I  =  6  [l  +  f  Q2],  and  Z:  =  5,  [l  +  f  (^)2],  (Vol.  I.  §  147). 
If  we  have  the  distance  e  between  the  suspension  rods,  their  num- 
ber for  a  length  BC  =  5,  is  n  =  -;  and  if  in  the  equation  x  =  ^—  a, 

we  substitute  for  y  the  values  o,  e,  2«,  3e,  4e,  &c.,  we  get  for  the 
lengths  of  the  suspension  rods: 


to  each  of  which  a  few  inches  are  to  be  added. 

From  the  weight  G  of  the  loaded  half  of  the  chain  AB,  the  hori- 
zontal tension  of  the  whole  chain: 

H  =  G  cotq.  a  =  —  G,  and  the  entire  tension  on  the  end  : 
S=_G_ 


sin.  tt       ^52  +  4a2 

If  we  know  the  modulus  of  strength  of  the  chains  and  suspension 
rods,  we  can  determine  the  sectional  dimensions  they  should  have. 
According  to  French  experience,  the  greatest  load  that  should  be 
brought  on  chains,  is  12  kilogrammes  per  square  millimetre  (or  about 
8  tons  on  the  square  inch),  and  for  cables  of  iron  wire  18  kilog.  per 
square  millim.,  or  about  12  tons  per  square  inch.  The  suspension 
rods  are  made  much  stronger  in  proportion,  as  they  have  to  resist  the 
shocks  of  loaded  wagons,  &c.,  passing  along  the  bridge.  The  load 
on  them  is  reduced  to  from  1|  to  3  tons  per  square  inch  of  section. 

§  35.  Sectional  Dimensions  of  the  Chains  and  Ropes. — In  order 
to  determine  the  dimensions  of  the  parts  of  a  suspension  bridge,  we 
have  to  take  into  consideration,  not  only  the  weight  of  the  road- way, 
but  also  the  greatest  weight  of  men,  as  troops,  or  of  cattle,  or  of 
wagons,  that  can  be  brought  to  bear  upon  it.  This  has  been  taken 
as  42  Ibs.  per  square  foot  of  surface  by  Navier,  but  in  the  case  of  a 
dense  crowd  of  persons,  it  might  amount  to  72  Ibs.  per  square  foot. 
Having  assumed  a  certain  maximum  load,  the  dimensions  of  the 
cross  and  longitudinal  beams  have  to  be  determined,  and  hence  we 
find  the  entire  weight  of  the  road-way.  If  we  put  the  sum  of  this 
constant  weight,  and  the  maximum  load  that  may  come  on  to  the 
bridge  =  Gv  and  the  modulus  of  strength  of  the  suspension  rods 

42 

=  K,  we  get  for  the  section  of  these  Fl  =  — l.     From  this  we  have 

K 


64  DIMENSIONS  OF  THE  CHAINS  AND  ROPES. 

the  weight  of  these  rods,  which  has  to  be  added  to  that  of  the  road 
and  load,  in  order  to  put  the  total  load  on  the  chain  G.  If  we 
put  the  section  of  the  chains  =  F,  and  the  specific  gravity  of  the 
iron  =  y,  we  have,  retaining  the  notation  as  above,  the  weight  of  the 
chains  : 


and  hence  the  total  load  on  one-half  the  bridge  : 

G=  G,  +  G2  =  G1+  Fb  [l  + 
and  the  strain  at  the  point  of  suspension 


sn.  a.  sn.  a 

But  for  the  necessary  security  S  =  FK  (where  K  is  the  modulus  of 
strength),  therefore  : 


+ 
i.  e.,  the  section  of  the  chains: 


Example.  The  dimensions  of  the  parts  of  a  suspension  bridge  of  150  feet  span,  15 
feet  deflexion,  and  25  feet  in  width  are  required.     Suppose  45  suspension  rods  on  each 
side,  we  have  then  44  equal  parts  of  3,409  feet  each.     The  length  of  these  rods,  com- 
mencing at  the  centre  would  be  0,  1L  =  0,031,  4  .  —  =  0,124,  9  .  —  —  0,279, 16 .  — 
22*  222  222  22* 

=  0,496,  25  .  —  =  0,775  feet,  &c.,  or  if  we  add  to  each  2  inches,  the  length  becomes : 

2,  2,37,  3,49,  5,35,  7,95,  11,30  inches,  &c. 

The  maximum  load  on  the  half  bridge,  we  shall  take  according  to  Navier 
75  X  25  X  42  Ibs.  =  78750  Ibs.,  and  if  the  road-way  weighs  a  little  less  than  a  ton  per 
foot  of  length  G,  =  157500,  and  the  section  of  all  the  rods  of  one-half  of  the  bridge: 

FI  = =72  square  inches.     The  whole  bridge  is  suspended  on  90  rods,  and, 

72     2 
hence  the  section  of  each  rod  is        '       =  1,6  square  inches,  or  the  diameter  of  the  rods 

must  be  1,427  inches.  According  to  the  rules  for  the  quadrature  of  the  parabola,  the 
mean  length  of  a  suspension  rod  =  $  that  of  the  largest,  therefore,  =  5  .15=5  feet, 
and  if  as  above,  we  add  2  inches  to  it,  then  it  =  5£  feet,  or  62  inches.  Thus  the  volume 
of  all  the  rods  is  90  X  62  X  1,6  =  8928  cubic  inches,  and  the  weight  taken  at  0,29  Ibs.  per 
cubic  inch  =  2598  Ibs.  The  half  of  this  added  to  the  above-found  weight  of  half  the 
road-way  gives  G=  158794  .  5  Ibs.,  and,  hence,  according  to  the  formula: 

F= ^L_ , 

if,  G,  =  158794,5,  K=  17500,  6  =  75  x  12  =  900,1  =  l£=s  0,2083 ,  y  =  0,29, 

6        72 

and««.a=        2a  3°  *  * 


17500  .  0,37 14  —  900 .  0,29  (1  +  f  .  0,2083')  ~~   6499,5  —  268,5  6231,0 


ELONGATION  OF  CHAINS.  65 

=  25,48  square  inches,  and,  therefore,  for  4  chains  the  section  of  each  would  be  6,37 
square  inches. 

§  36.  Elongation  of  Chains.  —  The  chains  are  elongated  by  the 
load,  and,  therefore,  the  deflexion  is  increased.  Changes  of  tem- 
perature also,  produce  variations  in  the  length  of  the  chains.  We 
must  know  the  effects  of  both  these.  If  the  deflexion  changes  from 
a  to  av  the  length. 


and  hence  the  elongation  of  the  chain  : 

,  =  7  —7  =  2  /<—  A  =  2  (ai  —  a)  (ai  +  a] 

M    b    )    3         b 

or  if  A  be  the  increase  in  the  deflexion,  and  if  we  put  as  an  approxi- 

mation a  +  <Zj  =  2  a,  Xj  =  |  _  A,  and,  therefore,  for  the  whole  chain 
b 

si  =  |  -  A,  inversely  A  =  f  -  a..      From  the  weight   G  of  the  half 
6  a 

bridge,  the  horizontal  tension  or  tension  at  the  apex,  H  =  G  cotg.  <*, 

and  the  tension  at  the  ends  :  S  =  --  ,  therefore,  the  mean  tension 

sin.  a 

=  H+S  =  G  (l  +  g°gij_;  and  the  extension  of  the  chains  caused 
2  2  sin.  o 

by  this  force  *  =  t1  +  f  os-  a)  .  _?_  .  2  I,  (Vol.  I.  §  183),  for  which 
2  sin.  a        FE 

2  Gb 
we  may  put  as  an  approximation  :  &  =  —  —  —  :  -  .     If  we  introduce 

this  value  into  that  for  A,  we  get  the  increase  in  the  deflexion  for 
the  loaded  chains  : 

A_3   b_    zab    ^f      a      >. 

a       FE  sin.  a          FE  sin.  a.     a  ' 

or  sin.  a  =  _      a          or  approximately  =  -  ,  we  get 
VV  +  4  a2  9. 

,      G      b3 
A-f;-F£-«T 

Malleable  iron  expands  0,0000122  of  its  length  for  a  rise  of  tem- 
perature of  one  degree  of  centigrade  (=,0000068  for  1°  Fahr.). 
This  increase  is,  therefore,  0,0000122  .  2  7  1  for  the  length  of  chain 
7,  and  a  rise  of  t  degrees  of  temperature,  or  0,0000244  7  1.  Putting 
this  in  the  expression  for  A,  we  get  the  increase  of  deflexion  for  a 
rise  of  temperature  t: 

A  =  f  .  -  .  0,0000244  .  It,  or  approximately  =  0,00000915  .  *  -. 
a  & 

In  like  manner  the  contraction  is  determined  for  decrease  of  tem- 
perature. 

Example.  Retaining  the  values  of  the  example  in  the  last  paragraph,  we  get  the 
increase  of  the  height  of  the  arc  corresponding  to  the  load,  taking  the  modulus  of  elas- 

6* 


66  PIERS  AND  ABUTMENTS. 

ticity  of  malleable  iron  =  29000000  (Vol.  I.  §  186),  and  adding  6841,8  Ibs.  the  half 
weight  of  the  chains  to  the  load  158794,5  Ibs. 

G  =  158794,5  -f  6841,8  =  165636,3  Ibs. 


=  g  .  ,__     9003        1397  .^      ^  &  ^       of  ratnw 

,  1>5,48  .  29000000      180'         739 

of  20°  C.  this  change  of  deflexion  is: 

0,00000915  .  20  .  ^2PI  =  0,4  inches. 
180 

Fig.  so.  §  37.  Piers  and  Abutments.  —  The  pro- 

portions of  the  piers  and  abutments  form 
an  important  consideration. 

If  S  and  Si  be  the  tension  on  the  ends  of 
the  chain,  Fig.  80,  and  o  and  Ol  the  angles 
of  inclination,  the  vertical  pressure  on  the 
pier: 

V2  =  V  +  Vl  =  S  sin.  0  +  Sl  sin.  «,, 
and  the  horizontal  pressure,  as  the  horizon- 
tal tensions  counteract  each  other, 

H2  =  H  —  Hi=  S  cos.  a  —  Si  cos.  or 

If,  now,  h  be  the  height,  b  the  breadth,  and  d  the  depth  or  thick- 
ness of  a  pier,  the  density  of  the  masonry  of  which  =  y,  its  weight 
is  b  d  h  y  =  G,  and  the  total  vertical  pressure  =  V2  +  G  =  S  sin.  a 
+  Si  sin.  otj  +  b  d  Ay.  In  order,  however,  that  the  horizontal  force  H2 
=  H  —  Hi  may  not  turn  the  pier  on  the  edge  B,  it  is  requisite  that 
the  statical  moment  : 

H2.LX  =  ff2h  =  (S  cos.  a  —  Si  cos.  Ol)  h 
should  be  less  than  the  statical  moment: 

(  yz  +  G)  B  L  =  (S  sin.  o  +  St  sin.  BI  +  b  d  h  y)  *  , 

i.  e.  it  is  requisite  that  : 

,2       /S  sin.  o  +  Si  sin.  «!  i  '      2  (S  cos.  a,  —  $,  cos.  oj 
d  hy  dy 


For  the  sake  of  security,  the  greatest  value  of  S  cos.  a  and  the 
least  value  of  Sr  cos.  ax  are  to  be  taken,  that  is  to  say  $..is  to  be 
taken  as  completely  loaded,  and  $,  as  unloaded.  This  formula  as- 
sumes that  the  forces  S  and  /S^  are  entirely  transferred  to  the  pier 
head,  which,  of  course,  only  takes  place  when  the  friction  on  the 
pier  head  exceeds  the  difference  S  —  ^  of  the  tensions.  According 
to  Vol.  I.  §  175,  this  friction  is: 


where  /  is  the  co-efficient  of  friction,  n  the  number  of  links  on  the 
pier  head,  and  j3  the  central  angle  corresponding  to  one  link,  it  is 
hence  requisite  that: 


|\H—  ll 


Sv 


PIERS  AND  ABUTMENTS. 


67 


S  <(l  +  2/«m.  |y&  Unless  this  condition  be  fulfilled,  the 
chain  will  slide  on  the  pier  head,  and  therefore  we  have  only  to  put 
S  =  (l  +  Zfsin.  l)'1^,  or  for  ropes  S  =  ef*  ^  (Vol.  I.  §  176), 

in  the  above  formula.  If  the  chain  or  cable  be  laid  upon  pulleys, 
this  difference  is  much  less,  and,  therefore,  the  requisite  thickness  of 
pier  is  less.  If  the  radius  of  the  pulleys  =  a,  and  the  radius  of  the 
axes  on  which  they  turn  =  r,  then: 

S^S,+f^(S  sin.  «  +  S,  sin.  oj, 
r 

for  the  friction  reduced  to  that  of  the  axis  may  be  put  =  /  -  (Ssin. 


a  +  Sl  sin.  Ol)  =  /  -  (  V  +  FJ.     If  the  rope  passes  over  rollers, 

then  the  friction  is  so  much  reduced,  that  we  may  put  S  =  S^ 

From  the  tension  S  on  the  land  or  back 
chains,  we  can  determine  the  dimensions  of  F'g-  81- 

the  retaining  wall  AC,  Fig.  81. 

The  strain  S  tends  to  turn  the  masonry 
J1C  round  C,  and  acts  with  a  leverage 
CJV  =  CD  sin.  a  =  I  sin.  o,  if  a  be  the 
angle  of  inclination  SDC  of  the  rope  to 
the  horizon,  and  I  the  length  CD  of  the 
wall.  The  height  of  the  wall  resists  with 
the  moment: 


G.CM= 


.-  =  ± 

2 


where  h  is  the  height  BC,  d  the  depth,  and 

y  the  weight  of  the  masonry.     For  equilibrium  S I  sin.  a  =  |  h  d  I2  y, 

and,  therefore,  the  requisite  width  of  wall  I  =  2  S  sm'  a .     To  insure 

stability  this  must  be  doubled.  That  such  a  wall  may  not  be  pushed 
forward,  the  friction/  (G —  S  sin.  a)  must  be  greater  than  the  hori- 
zontal force  S  cos.  o,  or,  /  G  »  S  (cos.  a  +  /  sin.  a), 

i.  e.  I  >  — _ -  (cos'  a  4-  sin.  a),  in  which  /may  be  taken  =  0,67. 
h  d  y  \    /  / 

Example.  For  the  suspension  bridge  mentioned  in  previous  paragraphs,  the  vertical 
force  of  the  loaded  chain:  V=  165636,3  Ibs.,  and  that  of  the  unloaded: 
V,  =  V—  78750  =  86886,3  Ibs.,  if  now  we  suppose  friction  pulleys  to  be  applied,  the 

radius  of  each  pulley  being  to  that  of  its  axis  as  —  =  i  and  /=  i,  the  friction  at  the  pul- 
leys would  be  J  .  }  .  (165636,3  -\-  86886,3)  =  15782,6  Ibs.,  or  much  less  than  the  dif- 
ference  of  the  tensions,  and  therefore  the  chains  would  move,  and  the  pulleys  turn  till 
the  tension  on  the  one  had  so  far  increased,  and  that  on  the  other  so  far  decreased  that 
the  difference  would  be  only  15782,6  Ibs.  If  now  the  height  of  the  pier  be  16  feet,  the 
thickness  4  feet,  and  the  weight  of  the  masonry  130  Ibs.  per  cubic  foot,  we  have  for  the 
necessary  width  of  piers : 


68  STRENGTH  OF  MATERIALS. 


252522'6        Aa.l5788.6«i.  ...  tfan  A,__  15782,6.  0,9285  _ 


16  .  4  .  130  4  .  130  260 

Therefore,  b  =  56'36  ~  ^  =  1,75  feet.     This  would,  in  practice,  be  made  4  to  5  feet. 

The  requisite  length  of  retaining  wall,  when  h  =  16  and  d  =  16  feet,  is: 

2  S  yirt.  «  __  2  .  165636,3         lg  g  which  would  bg  made  2Q  tQ  2&  -n          tice 

hdy  16.10.130 


STRENGTH  OF  MATERIALS.* 

The  strength  of  an  engineer's  work  depends  upon  its  proportions, 
the  materials  of  which  it  is  composed,  and  the  manner  of  putting 
them  together. 

As  to  stability,  a  structure  may  yield,  under  the  pressures  to  which 
it  is  subjected,  either  by  the  slipping  of  certain  of  its  surfaces  of  con- 
tact upon  one  another,  or  by  their  turning  over  upon  the  edges  of  one 
another.  The  former  case  very  rarely  occurs. 

The  strength  of  materials  depends  upon  their  physical  constitu- 
tion, viz :  form,  texture,  hardness,  elasticity,  and  ductility.^  The 
resistance  of  materials  in  buildings  is  tested  in  reference  to  various 
strains — compression — extension — detrusion — deflexion  under  a  cross 
strain,  and  fracture  under  a  cross  strain. 

A.  Compression. — In  prismatic  pieces  of  stone,  wood,  or  cast  iron, 
which  absolutely  crush  under  a  strain,  the  strength  is  directly  pro- 
portional to  the  transverse  area  of  the  piece. 

Pieces  exposed  to  compression  are  not  fairly  crushed,  but  in  some 
measure  broken  across,  where  their  height  is  to  their  diameter  or 
least  lateral  dimensions  in  the  case  of, 

Stone,  more  than  as  6     to  1? 
Wood,          "          "4     to  1 
Cast  iron,    "  "  3|  to  1 

Wrought  iron,        "  2J  to  1 

The  manner  in  which  materials  yield  under  a  crushing  strain  is 
very  remarkable,  as  is  exhibited  by  the  experiments  of  Rondelet, 
Vicat,  and  E.  Hodgkinson,  the  latter  of  whom  has  found,  that  the 
plane  of  rupture  is  always  inclined  at  the  same  angle  to  the  base  of 
the  column,  when  its  height  is  within  the  limits  above  mentioned. 
The  angle  of  rupture  depends  upon  the  nature  of  the  material.  In 
cast  iron,  for  instance,  it  varies  from  48°  to  58°  in  different  makes 
of  iron,  though  confined  to  narrow  limits  for  different  prisms  of  the 
same  make." — See  "Report  British  Association,"  1836,  and  Mose- 
ley's  "Engineering,"  p.  550. 

*  Professor  Weisbach  has  treated  this  subject  as  it  is  usually  given  in  elementary 
works  on  mechanics.  Excepting  as  exhibiting  approximately  the  laws  of  the  phenomena, 
the  "  theory  of  the  strength  of  the  materials"  has  many  practical  defects.  These  we  shall 
not  here  enumerate;  but  have  put  together,  in  as  concise  a  form  as  possible,  what  we 
consider  to  be  the  most  valuable  part  of  our  present  knowledge  on  this  subject  to  engi- 
neers or  architects  engaged  in  the  execution  of  works. 

t  See,  on  this  subject,  Poncelet's  "  Mecanique  Industrielle." 


STRENGTH  OF  PILLARS. 


69 


TABLE  OF  THE  RESISTANCE  OF  MATERIALS  TO  CRUSHING. 


Ibs.  per  sq.  inch. 

Ibs.  per  sq.  inch. 

Granite,  Scotch 

10804   to    8184 

n  ,         <  unseasoned 

.     6480 

"    .     Cornwall 

6292 

Oak'       I  seasoned 

10000 

Sandstone,  Dundee 

6490 

Mahogany 

8198 

"           Derby 

3110 

T  arfVi      5  unseasoned 

3200 

Marble  (white) 

9583 

Larch    I  seasoned 

5568 

Limestone  (Portland) 

6550 

„     ,       C  unseasoned 

3100 

Stourbridge  brick 

1695 

f*&*  I  seasoned 

5100 

n    ,       5  unseasoned     - 

6780         .     j 

Cast  iron,  good  common 

109800 

Deal       I  seasoned 

7290 

"      ,"    Stirling's  toughened    145500 

TWfh    $  ^seasoned 

7730 

Wrought  iron       .         .         .     56000? 

J3eech    >  seasoned 

93GO 

The  effect  of  seasoning  or  drying  timber,  in  increasing  its  strength, 
is  never  to  be  lost  sight  of.  In  wrought  iron,  a  strain  of  28000  Ibs. 
reduces  the  length,  and  causes  a  slight  lateral  bulging,  corresponding 
to  the  slight  reduction  in  length;  that  is  to  say,  for  a  compressive 
strain  of  about  f  ths  of  the  absolute  crushing-strain,  wrought  iron  is 
quite  "  crippled." 

Stirling's  process  of  toughening  cast  iron,  consists  in  adding  to  it 
proportions  of  malleable  scrap,  varying  according  to  the  nature  of 
the  cast  iron  in  its  normal  state. 

Scotch  hot  blast,  No.  1,  will  take  28  to  30  Ibs.  of  scrap  per  cent. 
a  a  a  ^Q  2  "  20  "  "  " 

Welsh  and  Staffordshire  hot  or  cold  blast  iron  require  a  less  ad- 
dition of  scrap. 

This  process  increases  the  strength  of  all  cast  irons,  from  50  to  80 
per  cent. 

The  strength  of  pieces,  such  as  pillars,  that  break  across,  but  are 
not  crushed  under  compression,  may  be  calculated  by  the  following 
formulas,  as  found  by  Mr.  Hodgkinson's  "  Experimental  Researches 
on  the  Strength  of  Pillars,"  published  in  the  Phil.  Trans.,  1840, 
and  in  his  edition  of  "Tredgold  on  Cast  Iron,"  published  1846. 

For  stone:  b  =  —  *     For  timber: 
P 

For  cast  iron.     Solid  pillar,  round  ends 


b  =—. 


"        "      flat  ends  b 

The  kngth  I  being  not  less  than  30  d. 

Hollow  pillars,  round  ends    b  =  a       '  ' 

The  length  1  being  not  less  than'lS  d. 

Hollow  pillars,  flat  ends        b  =  a ! — ~ — !_!. 

The  kngth  1  being  not  less  than  30  d. 

Wrought  iron,  round  ends     b  s=        ' 


flat  ends 


When  the  length  is  from  30  to  90  times  the  diameter. 


70  STRENGTH  OF  PILLARS. 

The  laws  indicated  by  the  formulas  do  not  hold  good  for  shorter 
columns. 


TABLE  OF  THE  VALUES  OF  a,  (  D  and  d  being  in  inches,  I  in  feet,  and 
the  result  b  being  the  crushing-weight  in  Ibs. 

Granite       .         .         flat  ends      .         .  25000? 

Sandstone  ......  15000? 

Marble        ......  24000? 

Dantzic  oak         .....  24542 

Red  deal     ......  17511 

Cast  iron  solid  pillar,  flat  ends      .         .  98922 

"      "       "         "     round  ends  .         .  33379 

Hollow  pillars,  flat  ends     .         .  99318 

"  "  round  ends          .  29074 

Wrought  iron  flat  ends     .         .  299617 

"  "  round  ends          .  95844 

The  numbers  here  given  are  co-efficients,  and  have  no  meaning, 
apart  from  the  special  position  they  occupy  in  the  formulas. 

In  all  pillars  of  cast  iron,  whose  length  is  thirty  times  the  diameter 
or  upwards,  the  strength  of  those  with  flat  ends  seems  to  be  three 
times  as  great  as  the  strength  of  those  of  the  same  dimensions  with 
rounded  ends  :  when  I  is  less  than  30  d,  the  ratio  of  the  strength  of 
pillars  of  the  same  dimensions  with  flat  and  with  rounded  ends,  is 
very  variable. 

When  pillars  are  reduced  in  length  below  the  proportion  above 
indicated,  there  is  a  falling  off  of  their  strength,  nearly  in  propor- 
tion to  the  reduction  in  the  length  of  the  pillar;  and  this  obviously 
must  be  the  case,  as  the  strength  to  resist  flexure^  under  a  compres- 
sive  strain,  increases  as  the  fourth  power  of  the  diameter,  whilst  the 
resistance  to  crushing  increases  only  as  the  square  of  the  diameter. 

For  pillars  of  less  length  than  15  times  their  diameter,  there  is  a 
falling  off  in  the  resistance,  on  account  of  the  change  produced  in 
the  position  of  the  molecules  of  the  material  by  the  great  weight 
necessary  to  break  them:  Mr.  Hodgkinson  has,  however,  given  a 
formula  which  includes  this  case,  and  by  which  the  strength  of  the 
pillars,  however  short,  may  be  deduced  from  the  results  of  the  for- 
mulas for  long  columns,  when  the  crushing  strength,  of  the  material 

is  known.     The  formula  is  y  =  —  C—  in  which  b  is  the  strength  of 


the  pillar,  as  calculated  by  the  rules  for  long  pillars,  and  c  the  crush- 
ing weight  of  the  material,  and  y  —  the  strength  of  the  short  pillar. 

In  similar  pillars,  the  strength  is  nearly  as  the  square  (1,865 
power)  of  the  diameter,  or  of  any  other  lineal  dimension;  and  as 
the  area  of  the  section  is  as  the  square  of  the  diameter,  the  strength 
is  nearly  as  the  area  of  the  transverse  section. 

The  strength  of  pillars  not  less  than  30  times  their  diameter: 
that  of  cast  iron  with  rounded  ends  being  set=  1000 


THE  TENSILE  STRAIN.  71 

Wrought  iron     .         .         .         .  is  =  1745 

Cast  steel           .         .         .         .  =2518 

Dantzic  oak,  square  ends    .         .  =  108,8 

Red  deal =    78,5 

In  all  long  pillars,  whose  ends  are  firmly  fixed,  the  power  to  resist 
breaking  is  equal  to  that  of  pillars  of  the  same  diameter  and  half 
the  length,  with  the  ends  rounded  or  turned,  so  that  the  strain  runs 
through  the  axis. 

B.  Extension. — When  a  tensile  strain  passes  up  the  centre  of  a 
piece  of  stone,  wood,  or  metal,  the  resistance  is  proportional  to  the 
transverse  area  of  the  piece. 

TABLE  OF  THE  RESISTANCE  OF  MATERIALS  TO  RUPTURE  BY 
TENSILE  STRAIN. 

Stone.     Portland     ....         857  Ibs.  per  sq.  inch. 
Fine  sandstone    .         .         .         215 
Brick          .         .         .         .         275  to  300 

Glass 3565 

Hydraulic  lime,  best         .         .         .         168 

Good 142 

Mean  quality  .....         100 
Common  lime  .....  43 

Timber.    Deal          .         .         .         .     12857  to  11549 
Beech        ....     17850 
Oak  ....       9198  to  12780 

Mahogany         .         .         .     16500 
Larch        ....       9700  to  10220 
Poplar       .....  7200 
Cast  iron  (Hodgkinson)    .         .         .     13505  to  17136 
"        (Rennie)  ....     19200 
(Cubitt)    ....     27773? 
"        Stirling's  toughened  .         .     28000 
Wrought  iron  bars  ....     65520  to  56000 
Wire  (hard)        .         .  128000  to  65360 
Wire  (annealed),  half  the  strength  of  hard. 
Plates         .         .         .     52100 
Brass  wire  (hard)     ....     98960  to  63000 

Annealed        .         .         .     49000 

Gun  metal  (hard)      ....     36368 

Copper  rolled  ....     35000 

"      cast     .         .         .         .         .     19200 

Ropes.     Hemp        .         .         .         .     1  ton  per  Ib.  weight  per 

fathom. 

Wire,  (Newall  and  Co.)       .     2  tons  per  Ib.  weight  per 

fathom. 

In  reference  to  the  above  table,  it  may  be  stated  that  it  contains 
numbers  which  are  the  mean  values  of  the  tensile  strain,  as  deduced 


72  THERMOTEXSIOX. 

after  a  careful  weeding  of  the  experimental  results  that  have  hitherto 
been  published. 

[  Thermotension^  or  the  Effect  of  Heat  on  the  Tenacity  of  Iron. — 
The  following  table  exhibits  the  effect  of  heat  on  the  tenacity  of  iron, 
both  while  actually  hot  and  also  subsequent  to  the  application  of  a 
strain  at  high  temperature.  The  comparisons  are  made  on  thirty- 
two  different  specimens  of  iron,  the  origin  of  which  is  designated  in 
the  first  column  of  the  table.  The  temperature  at  which  either  the 
"hot  fracture"  or  the  hot  strain  was  made  on  each  bar,  and  which 
produced  the  strengthening  effect  of  "thermotension,"  is  contained 
in  the  second  column.  The  third  contains  the  number  of  trials 
made  on  each  specimen  of  iron  to  ascertain  its  strength  in  its  ordi- 
nary state  and  temperature,  as  it  came  from  the  hammer  or  the 
rolls,  and  before  being  put  under  strain  at  a  high  temperature. 
Column  fourth  shows  the  number  of  times  the  specimen  was  broken, 
or  at  least  strained,  at  the  temperature  marked  in  column  third. 
Column  fifth  gives  the  number  of  fractures  made  on  the  specimen  to 
obtain  the  average  strength  after  being  heated,  strained,  and  then 
cooled  again  to  ordinary  temperature.  Columns  six,  seven,  and 
eight,  contain  the  absolute  strength  given  in  the  three  different 
states  respectively.  Column  nine  exhibits  the  per  centage  increase 
of  strength  by  treatment  with  thermotension,  and  ten,  the  difference 
in  strength  between  the  iron  at  ordinary  temperature  in  its  original 
state,  and  that  which  it  possessed  while  heated  as  in  column  third. 
In  three  cases  only  does  it  appear  that  the  strength  had  been  dimi- 
nished by  heating  up  to  the  point  at  which  the  trials  were  made. 
One  of  those  trials  was  at  766°,  one  at  662°,  and  the  third  at  552°. 
The  average  temperature  at  which  the  effect  was  produced  was 
573.7°,  at  which  point  the  tenth  column  shows  that  the  strength  of 
thirty  varieties  of  iron,  was  5.9  per  cent,  greater  than  at  ordinary 
temperatures,  say  at  60  or  80  degrees. 

It  also  appears  that  the  average  gain  of  tenacity  in  thirty-two 
samples  of  iron,  by  the  process  above  mentioned,  was  17.85  per 
cent.,  ranging  from  8.2  to .28. 2  per  cent.  In  a  report  by  the  Editor 
to  the  Bureau  of  construction,  equipment,  and  repairs  of  the  Navy 
Department  of  the  United  States,  it  is  proved  that  the  average  gain 
of  length  of  bolts  of  iron  treated  at  the  Washington  Navy  Yard,  by 
this  same  process,  was  5.75  per  cent.,  and  the  gain  of  strength  16.64, 
making  together  the  gain  of  value  22.4  per  cent.  The  addition  of 
5.75  to  17.85,  gives  23.6  per  cent,  for  the  total  gain  of  value.  In 
many  instances  the  experiments  proved  the  gain  of  length  to  exceed 
7  per  cent.  The  total  elongation  of  a  bar  of  iron,  broken  in  its 
original  cold  state,  is  from  two  to  three  times  as  great  as  the  same 
force  would  produce  upon  it  if  applied  at  a  temperature  of  573°, 
which  force  will,  moreover,  not  break  the  bar  at  that  temperature. 


THERMOTENS10N. 


73 


TABLE  EXHIBITING  THE  EFFECT  OF  HEAT  ON  THIRTY-TWO  VARIETIES 
OF  MALLEABLE  IRON. 


1  1 

•a 

I- 

.SS,; 

1 

S| 

11    . 

IP 

<z 

c 

•^ 

ca  -^ 

"r  "-  2 

•— 

fE 

x=  5 

gl  If 

3 

«  '5 

5s  | 

M 

el 

Hi 

Iff 

OF  IRON  TRIED. 

2  >  J5  *  ;  S 

'•a 

!si 

« 

•5  - 

Is 

ejSag 

P 

of-£_. 

'o  r^s 

!._  J"S 

Ifll 

&•-=  z~° 

0 

o-a 

>.=! 

£  « 

>*| 

*   —  JC 

g.|** 

Pj    S 

ja 

o  o 

£ 

»l 

°i'I 

'ill 

Sa'isbury  (Conn.),  gun  bar 

5540 

4 

5 

59.271 

60459 

65090 

8.2 

+  2.0 

Maramec  (Mo  ),  bar  iron 

528 

7 

3 

53775 

54.273 

59044 

9.8 

+  0.9 

Phillipsburgh(Pa),wire 

500 

4 

4 

79.720 

80.488 

841.87 

104 

+  0.9 

Ellicott's  Baltimore  boiler  plate 

770 

3 

5 

56644 

56,644 

63.132 

10.7 

+  0.0 

•  (                         it                         tl              U 

662 

4 

7 

58.8!)9 

58.181 

64,820 

109 

~  1  2 

Salisbury  (Conn.),  gun  bar 

550 

3 

7 

59654 

60,323 

66.638 

117 

+  1-1 

«              u              u 

590 

4 

9 

59  032 

62.952 

67.384 

13.5 

+  66 

Swedish  bar  iron 

530 

3 

58.012 

59.775 

66.»34 

14.3 

4-30 

Nashville  (Tenn.).  bar  iron 

520 

7 

5 

54,934 

58451 

62.600 

145 

+  6.4 

Salisbury  (Conn  ),  gun  bar 
Ellicott's  Baltimore  forged  bar 
Spang  &  Son,*  hammered  plate 
Blake  &  Co..*  hammered  plate 
Salisbury  (Conn.),  gun  bar 

572 
394 
766 
572 
580 

5 
1 
3 
6 

4 

10 
1 
1 
4 
5 

58305 
57.182 
57664 
60532 
55977 

58.895 
63,322 
54.819 
62,278 

63.558 
65.960 
66500 
66,941 
65,883 

15.0 
15.3 
15.8 
16.3 
17.0 

+  1.0 
+10.7 
—  4.9 
+  2.8 
no  hot  frac. 

«                  tt                  a 

564        4 

8 

54,644 

60.215 

64363 

17.6 

+10.2 

a                 «                 u 

5 

5 

58299 

64278 

68.988 

18.4 

+10.0 

"                         "                         U 

630 

4 

6 

57.433 

60,010 

67.569 

187 

+  4.5 

English  "best-  best"  cable  bolt 
Spang's  Pittsburgh  humm'd  plate 
Nashville  (Tenn.),  bar  iron 

560 
552 
680 

3 
7 

10 
3 
6 

62.466 
56762 
52  729 

55932 
58.534 

71,000 
62,736 
62,127 

19.3 
194 
195 

no  hot  frac. 
-  1.4 
+11.0 

Mason  &  Miltenberger,*  piled 

574 

4 

2 

55426 

60.083 

68,a39 

195 

+  8.4 

Nashville  (Tenn.).  bar  iron 

562 

4 

2 

52.194 

59.623 

62,433 

19.8 

+142 

Silicon's  Baltimore  hoiler  iron 

553 

3 

61.519 

66.450 

73.898 

201 

+  80 

Schaenberger's  Pittsburgh  boiler 
Maramec  (Mo  ),  bar  iron 

630 
564 

1 
5 

3 
9 

53.803 
49.974 

56.159 
52.158 

64.926 
58.126 

20.6 
206 

+  4.4 
+  4.3 

Nashville  (Tenn.),  bar 

578 

5 

5      52.406 

59.192 

62,951 

21.1 

--12  9 

Russia  sable  bar 

5S4 

5 

3      76.071 

77161 

92470 

215 

-•14 

Maramec  (Mo.),  bar 

576 

6 

4 

43386 

50.067 

53.368 

230 

-  -15.4 

Ellicott's  hammered  bar 

394 

1 

1 

53.176 

56.570 

69767 

258 

--  6.4 

Salisbury  (Conn.)  gun  bar 
Maramec  (Mo),  bar 

575 
574 

3 

5 

6 
4 

52873 

45.586 

60.988 
51,437 

66,685 
58,252 

266 
281 

-  -15.3 
-  -12.8 

Blake's  Pittsburgh  hamm'd  plate 

564 

6 

4 

52937 

58,284 

65.425 

282 

--101 

Mean, 

573.7|  129 

36  ;153 

Mean, 

17.85 

+  5.9 

Fig.  82  represents  the  tenacity  of  wrought  iron  at  various  tem- 
peratures from  0°  up  to  1317°  as  measured  in  parts  of  the  total 
maximum  tenacity,  the  line  a  b  representing  that  maximum,  and  the 
line  0°d  (indefinite  towards  d)  being  the  scale  of  observed  tempera- 
tures, in  degrees  Fahrenheit  marked  below  it.  The  vertical  dotted 
lines,  or  ordinates  of  the  curve,  therefore,  exhibit  temperatures,  and 
the  corresponding  horizontal  ones,  or  abscissas,  show  diminutions  from 
the  maximum  strength,  at  the  temperature  observed.  Thus,  at  a 
temperature  of  1030°,  the  diminution  from  maximum  tenacity  is 
.4478,  and,  consequently,  the  remaining  strength  is  55.22  per  cent. 
At  1187°  the  diminution  is  .6352,  and  the  remaining  strength  36.48 
per  cent.,  and  at  1245°  (a  dull  red  heat  in  daylight)  the  diminution 
is  .6715,  and  the  remaining  cohesion  only  32.85  per  cent.,  &c. 

*  Of  Pittsburgh. 


VOL.  II. — 7 


74 


THERMOTENSION. 


Fig.  82. 
Maximum  tenacity  of  iron. 


THE  TENSILE  STKAIN.  75 

•For  a  more  full  exposition  of  the  effect  of  heat  on  the  tenacity  of 
iron  under  direct  tension,  and  for  investigations  of  the  relation 
between  temperature  and  tenacity,  reference  may  be  had  to  the 
"Report  on  the  Strength  of  Materials  for  Steam  Boilers,"  page 
212—218. 

At  page  75  of  the  same  report,  will  be  found  the  law  of  tenacity 
as  affected  by  temperature  for  rolled  copper.  In  that  metal  no  in- 
crease of  strength  takes  place  from  increase  of  temperature  in  any 
part  of  the  scale;  and  the  law  eliminated  from  about  180  comparisons 
of  different  experiments  on  several  specimens  of  copper,  is,  that  the 
diminutions  of  strength  by  augmentations  of  temperature  follow  the 
principle  of  a  parabola,  of  which  the  ordinates  representing  the  ele- 
vation of  the  temperature  above  32°,  have  to  the  abscissas  repre- 
senting the  diminutions  of  tenacity,  a  relation  expressed  by  saying, 
that  the  third  powers  of  the  temperature  are  proportional  to  the  second 
powers,  of  the  diminution  of  strength  which  they  produce.  This  law 
was  ascertained  in  the  following  manner :  Putting  t  =  any  observed 
temperature  above  32° ;  t1  =  any  other  observed  temperature  above 
the  same  point ;  d  =  the  diminution  of  tenacity  by  the  former  tem- 
perature and  d'  =  that  by  the  latter :  also  making  x  =  that  power  of 
the  temperature  according  to  which  the  diminution  of  tenacity  takes 

tx       d1 
place;  we  have,  by  the  supposition  tx  :  t'x  :  :  d  :  d',  or— ^  =  — . 

From  this  we  derive  the  expression  Xgsikff.  d'  —  log.  d^ 

log.  t'—log.  t 

Example.  At  a  temperature  of  1016°  the  tenacity  of  a  bar  of  copper  was  found  to  have 
been  diminished  66.91  per  cent,  below  its  strength  at  32° ;  at  the  temperature  of  492° 
it  was  21.33  per  cent,  below  what  it  was  at  32°;  according  to  what  power  of  the  tem- 
perature did  the  tenacity  vary? 

Here*  = ^669 1  -  fog.  .2133 =  ^  ,  „  .  „„  ..d.  f   ^ 

log.  (1016  —  32)  — fog.  (492  —  32) 
t>  :  f  »  :  :  cP  :  <f\ 

Transforming  this  into  an  equation,  we  get   ( —]    —  |_),and_  . —  I      j7,  or   df 

\t  /  \d  /  d          \t  / 

=  d  (  —  J*.  From  this  £  (log.  t'  —  fog.  t )  -f-  fog.  d  =  log.  df;  by  which,  knowing  the  dimi- 
nution d  at  any  one  temperature  t,  we  are  enabled  to  calculate  what  it  will  be  at  the 
temperature  t'.] 

In  reference  to  cast  iron,  the  first  or  lower  numbers  (p.  70)  are  the 
results  of  Mr.  Hodgkinson's  experiments  ;  the  higher  number  is  the 
result  of  numerous  experiments  made  for  Mr.  Thomas  Cubitt  by  Mr. 
Dines.*  This  difference  is  chiefly  of  importance  in  respect  of  there 
being  a  discrepancy  so  wide,  between  results  stated  by  two  careful 
experimenters.  In  reference  to  the  experiments  on  Mr.  Morries 
Stirling's  toughened  iron,  they  were  made  by  the  same  direct  means 
as  were  all  Mr.  Hodgkinson's  experiments.  The  tensile  strain  of  cast 
iron  is  seldom  brought  directly  into  action;  and  the  part  it  plays  in  the 
resistance  to  cross  strains  is  evidently  not  that  for  which  the  direct 
strength  shown  by  Mr.  Cubitt's  experiments  can  be  attributed  to  it. 

*  See  Mr.  Henry  Law's  edition  of  Gregory's  "Mathematics  for  Practical  Men,"  p.  375. 


76 


THE  TENSILE  STRAIN. 


The  elongation  of  wrought  iron,  under  a  given  tensile  strain,  may 
be  judged  of  from  the  following  experiment.* 


Load  per  square  inch  in  producing  an  elongation  of 

Load  per 
square  inch, 
producing 
fracture. 

Total  elonga- 
tion divided 
by   original 
length. 

?flff 

TTiTI 

* 

Jv 

Ibs. 
34700 

Ibs. 
40980 

Ibs. 
46124 

Ibs. 
52122 

Ibs. 
56834 

Ibs. 
.086 

According  to  Vicat,  the  elongation  of  iron  wire  for  a  load  of  1428 
Ibs.  per  square  inch,  or  J0  the  breaking  strain,  amounts  to  0,000(^57. 

Mr.  E.  Hodgkinson's  experiments  have  proved,  in  like  manner, 
that  no  material  is  so  elastic  as  to  recover  itself  perfectly  from  even 
very  small  loads  allowed  to  act  for  a  considerable  time,  and  the  de- 
fect of  elasticity  is  nearly  as  the  square  of  the  weight  applied. 

The  modulus  or  co-efficient  of  elasticity,  is  a  term  first  suggested 
by  Dr.  Thomas  Young,  to  denote  the  measure  of  the  elastic  reaction, 
or  the  energy  of  the  resistance  of  any  substance,  and  is  represented 

thus:  £=-^r. 
Ji  i 

Where  E  is  the  co-efficient  of  elasticity,  P  the  weight  in  pounds, 
producing  the  proportional  elongation  i  ( = where  Z=  the  elonga- 
tion, and  L  the  original  length)  in  a  bar  with  a  base  of  sectional 
area  A. 

-ri  a 

Rigidity  is  expressed  by  the  ratio . 

Ju 

Thus,  the  elastic  resistance  of  a  prism  of  any  material,  is  really 
only  the  rigidity  referred  to  the  unit  of  length  of  the  prism. 

[*  In  the  report  of  the  Committee  of  the  Franklin  Institute,  on  the  materials  for  steam 
boilers,  p.  219-20.  will  be  found  very  numerous  observations  on  the  elasticity  of  iron,  of 
which  the  following  may  be  cited  as  the  results  of  direct  measurement. 


Bar   49  Boiler 
plate  from  Juni- 
ata  blooms. 

Recoil,  when  relieved  from  strain, 
in  parts  of  original  length. 

Breaking  weight 
in  Ibs.  per  square 
inch. 

57.565 

Total  elongation 
after  frac  ture. 

6.9  per  cent. 

Sffff 

51.030  Ibs. 
per  square  inch. 

54.800  Ibs. 
per  square  inch. 

Bar  226. 

si* 
43.SOO  Ibs. 
per  square  inch. 

49.053 

6.25  per  cent. 

rh 

Bar  228.                      34.804  Ibs. 
1  per  square  inch. 

40.643 

Bar  230. 

47.155  Ibs. 
per  square  inch. 

49.368 

AM.  Ed.] 


THE  ELASTIC  RESISTANCE. 


77 


TABLE   OF   DATA   CONNECTED  WITH   THE   ELASTIC  RESISTANCE  OF 
MATERIALS. 


!«£ 

"is  • 

.S  a 

Ill 

1  1  i!  t 

|  o| 

3    ^* 

Name  of  material. 

M   -S     O 

o  'S  °  J 

^  ~  5 

111 

III* 

ill 

11 

^i    « 

Oak         '       . 

2856 

0.00167 

0.23 

Ibs. 
1,713600 

Yellow  pine    

3332 

000117 

0.33 

1,856400 

Red  pine         

4498 

0.00210 

0.44 

•2,142000 

Larch      .          ...... 

2470 

0.00192 

0.30 

1,285200 

Beech      

3355 

0.00242 

0.30 

1,385160 

Bar  iron,  ordinary  qualjjy 

17,600 

0  00062 

030 

•28,400000 

"       "      Swedish  hammered  )      . 
"       "      English  rolled             }  sele° 

24,400 
18,850 

0.00093 
0.00072 

0.44 
0.37 

29,365000 
29,465000 

Wire,  No.  9,  unannealed 

47,532 

0.00165 

0.49 

28,825000 

"            "     annealed     .... 

36,300 

0.00129 

0.58 

28,081000 

Steel  plates,  tempered  blue 

93,720 

0.00222 

0.67 

42,600000 

Steel  wire  of  commerce  .... 

35,700 

0.00120 

0.50 

29,500000 

Cast  iron         ...... 

17,000000 

to 

13,000000 

NOTE. — By  "Limit  of  Elasticity,"  is  meant  the  limits  within  which  displacement  of 
the  parts  of  materials  under  strain  may  be  called  into  play  without  permanent  palpable 
derangement,  or  crippling. 

C.  Detrusion  is  the  resistance  that  the  coherence  of  the  particles 
of  materials  opposes  to  their  sliding  on  each  other,  under  a  detrusive 
strain. 

The  resistance  to  detrusion,  or  the  "  force  necessary  to  shear 
across"  any  material,  is  called  into  play  at  the  joints,  and  in  the 
bolts  of  framings  of  timber  and  iron,  and  the  rivets  of  steam 
boilers,  &c. 

The  resistance  of  deal  to  detrusion  in  the  direction  of  fibre  is  592 
Ibs.  per  square  inch. 

The  resistance  of  cast  iron  to  detrusion  is  about  73000  Ibs.  per 
square  inch,  as  deduced  from  experiments  on  crushing. 

The  resistance  of  wrought  iron  to  detrusion,  or  to  a  force  "  shear- 
ing it  across"  is  45000  to  50000  Ibs.  per  square  inch,  or  from  70  to 
80  per  cent,  of  the  resistance  to  a  direct  tensile  strain. 

D.  Deflexion. — When  a  beam  is  deflected  by  a  cross  strain,  the 
side  of  the  beam  which  is  bounded  by  the  concave  surface  is  com- 
pressed, and  that  bounded  by  the  convex  surface  is  extended.     The 
surface  at  which  extension  terminates  and  compression  begins,  is 
termed  the  neutral  surface. 

The  property  of  elasticity,  inherent  in  all  substances  in  a  greater 
or  less  degree,  causes  them  to  resume  their  original  form,  very 
nearly,  when,  under  forces  of  compression,  extension,  or  deflexion, 
they  have  undergone  a  limited  change  of  form.  Up  to  this  limit, 
the  amounts  of  extension  and  compression  for  a  given  cross  strain 

7* 


78  ON  FRACTURE. 

are  nearly  equal,  and,  therefore,  the  neutral  surface  lies  very  nearly, 
if  not  accurately,  in  the  centre  of  gravity  of  the  cross  section  of  the 
beam. 

Beyond  this  limit  the  position  of  the  neutral  surface  changes,  as 
the  flexure  increases ;  because,  in  stone  and  cast  iron  at  least,  the 
resistance  to  compression  is  greater  than  the  resistance  to  extension, 
whilst  the  amount  of  deformation,  under  the  compressive  strain,  is 
less  than  under  an  equal  tensile  strain.  In  wrought  iron,  as  it  pos- 
sesses great  ductility,  this  limit  occurs  much  later  than  in  cast  iron. 
In  timber,  the  resistance  to  extension  is  greater  than  that  to  com- 
pression, and  its  want  of  homogeneity  renders  the  limit  alluded  to 
very  variable. 

The  general  law  of  deflexion  is,  that  it  increases,  cseteris  paribus, 
directly  as  the  cube  of  the  length  of  the  piece,  and  inversely  as  the 
breadth  and  cube  of  the  depth.*  , 

E.  Fracture. — The  theory  of  deflexion,  which  gives  the  displace- 
ments of  the  parts  of  beams  before  the  conditions  even  of  crippling, 
has  few  practical  applications ;  while  equations  for  the  resistance  to 
fracture,  which  is  what  is  essential  in  practice  to  be  known,  are  more 
simply  established. 

The  hypothesis  for  the  theory  of  resistance  of  materials  to  fracture 
or  rupture,  propounded  by  Galileo,  consists  in  placing  the  horizontal 
axis  of  equilibrium  at  the  lowest  point  of  the  section  of  rupture,  or 
in  supposing  the  material  incompressible ;  and  he  considered  the 
internal  force  developed  at  each  point  of  the  section  as  constant  for 
every  point. 

The  hypothesis  commonly  attributed  to  Mariotte  and  Leibnitz, 
consists  in  like  manner  in  placing  the  horizontal  axis  of  equilibrium 
at  the  lowest  point  of  the  section,  and  in  supposing  the  internal  force 
developed  at  each  point  proportional  to  the  distance  of  that  point 
from  the  axis  of  equilibrium. 

The  hypothesis  now  generally  adopted,  consists  in  admitting  that 
the  resistance  of  each  point,  at  the  instant  rupture  is  going  to  take 
place,  continues  proportional  to  the  extension  and  compression,  and, 
therefore,  that  the  axis  of  equilibrium,  or  neutral  surface,  has  the 
same  position  as  in  the  case  of  a  very  small  deflexion. 

Experiments  have  proved  that  none  of  these  hypotheses  is  true, 
and,  that,  according  to  the  physical  constitution  of  the  material,  the 
formula  deduced  from  the  one  or  the  other  may  be  taken  as  repre- 
senting experiments.  Experiments  on  cast  iron  are  best  represented 
by  the  deduction  from  Galileo's  hypothesis ;  those  on  stone,  by 
Marietta's,  and  those  on  timber  and  wrought  iron,  by  the  modern 
hypothesis,  announced  by  Hooke,  and  first  developed  by  Dr.  T. 
Young. 

The  formula  commonly  employed  for  reducing  experiments,  or 
for  calculating  dimensions  by  aid  of  experiments,  on  beams  of  uni- 

*  For  the  most  complete  development  of  this  subject,  the  student  is  referred  to  Mr. 
Moseley's  work,  "Engineering  and  Architecture,''  Part  V. 


ON  FRACTURE. 


79 


form  rectangular  section,  fixed  at  one  end  and  loaded  at  the  other, 

is  r  _•£**. 

nl 

On  Galileo's  hypothesis      .         .         .         .     n  =  2 
On  Leibnitz  and  Mariotte's         .         .         .     n  =  3 
On  Young's  hypothesis       .         .         .         .     n  =  6 
The  mean  of  experiments  gives  for  cast  iron     n  =  2.63 
"  stone  .         .         .     n  =  3 

"  "     wrought  iron  and  wood     n  =  6  ? 

To  answer  the  imperfection  of  the  theory,  however,  f1  is  substi- 
tuted for  /;  or  for  the  resistance  to  a  direct  tensile  or  compressive 
strain  there  is  substituted  a  co-efficient  of  the  composite  resistance 
to  fracture,  under  a  cross  strain. 

The  most  convenient  general  formula  in  use  for  calculating  the 

f1  I 

resistance  to  fracture  under  a  cross  strain  is  W  =  J-  —  . 

/q 

Where  W=  the  breaking  weight,  1=  the  moment  of  inertia  of 
the  cross  section  of  the  beam,  round  an  axis  passing  through  its 
centre  of  gravity,  c1  the  distance  of  the  neutral  surface,  from  the  side 
at  which  the  material  gives  way  ;  and  I  the  length.  The  beam  is 
supposed  fixed  in  the  circumstances  above  mentioned. 

For  a  beam  supported  at  each  end,  and  loaded  in  the  middle, 

this  becomes  W  =  —  —  ,  and   for   beams    of  triangular  section  : 


For  a  beam  supported  at  each  end,  if  the  load  be  uniformly  dis- 

Sfl  I 
tributed  over  it,  we  have  W  =  —  -  ,  and  for  beams  of  rectangular 


section,    W  =  - 
o 


—  - 
I 


If  the  weight  of  the  beam  G  be  taken  into  account,  the  above  for- 


mulse become  respectively  W+  |  Q-  =  -j—,  and  W+  &  = 


8/1! 


For  the  forms  of  transverse  section  commonly  met  with  in  prac- 
tice, the  values  of  I  in  terms  of  the  breadth  6,  and  depth  rf,  of  the 
beam,  are  as  follows: 


1.  Rectangular  section. 

2.  Circular  section. 

3.  7  shaped  and  hollow  rect- 
angular, 6,  and  dv  being    the 
breadth  and  depth  of  hollow. 

4.  Hollow  cylinder,  or  annu- 
lar section,  r^  =  radius  of  hol- 
low. 

5.  Inverted  JL  (Mr.  Hodgkin- 
son's  for  cast  iron).     When  *4, 
Az  *#3  are  the  areas,  and  d,  d,  d, 
the  depths  of  the  top  flange,  the 
bottom  flange  and  the  uniting 
rib  respectively. 


(rf,  —  </8)  *^,  — 


c,  depending 
on  the  form 
of  the  beam. 


80 


MODULUS  OF  RUPTURE. 


The  following  table  contains  values  of  f1,  or  modulus  of  rupture, 
being  deductions  from  experiment  by  the  formula  fl  =  Q  ,  all 
dimensions,  that  is,  ?,  b,  and  d,  being  in  inches. 


Name  of  material. 

Stone  (Rochdale)    . 

"     Yorkshire  flag 

"     Caithness  slate      . 

Beech 

Birch 

Deal  (Christiania)  . 

"    Memel  . 

Fir 

Larch    . 
Oak,  English 

"    Dantzic 
Cast  iron        . 

"        Hot  blast  mean 
"        Cold  blast  mean 

•  "        Stirling's  toughened 
Wrought  iron 


Modulus  of  rupture. 
Ibs. 

.  2358 
.  1116 
.  5142 
.  9336 
.  9624 
.  9864 
.  10386 
.  6700 
.  6894 
.  10000 
.  8742 
30000  to  46900 
.  36900 
.  39987 
.  46750 
.  54000 


Working  load. 
Ibs. 

235 

112 
514 

1550 
1600 
1640 
1700 
1100 
1150 
1700 
1500 

5000  to  8000 
6000 
6500 
7800 
9000 


The  following  table,  drawn  up  by  Mr.  Hodgkinson,  gives  the  re- 
lation between  the  resistances  to  crushing,  rupture  by  tension,  and 
by  cross  strain. 


Material. 

Assumed  resistance 
to  crushing  per 
square  inch. 

Mean  resistance  to 
rupture  by  extension 
per  square  inch. 

Mean  transverse 
strength  of  a  bar,  1 
inch  square  and  1  foot 
long. 

Timber  .     .     . 
Cast  iron     .     . 
Stone     .     .     . 
Glass      .     .     . 

1000  or  1 
1000  or  1 
1000  or  1 
1000  or  1 

1900  or  1.9 
158  or  0.16 
100  or  0.1 
123  or  0.125 

85.1  or  0.045 
19.8  or  0.02 
9.8  or  0.01 
10.    or  0.01 

From  this  table  we  get  an  idea  of  the  extent  to  which  the  mutual 
dependency  of  the  fibres  or  particles  of  the  material  comes  into  play 
when  the  pieces  are  bent. 

This  table  indicates,  too,  that  the  resistance  of  the  same  area  of 
cross  section  must  vary  according  to  the  disposition  of  the  material 
compressed  and  extended  in  the  section.  Mr.  Hodgkinson  has 
proved,  in  reference  to  this,  that  for  cast  iron,  one  mode  of  dispos- 
ing the  iron  in  the  section  gives  a  greater  strength  per  square  inch 
of  the  section  than  another,  in  the  ratio  of  40  to  23,  and  the  prin- 
ciple holds  in  other  materials. 

For  the  inverted  ^-shaped  girder,  the  strongest  form  is  that  in 


TUBULAR  BRIDGES.  81 

which  the  bottom  flange  is  six  times  the  area  of  the  top  flange. 
When,  in  these  girders,  the  length,  depth,  and  top  flange  are  con- 
stant, and  the  thickness  of  the  vertical  rib  between  the  flanges  small 
and  constant,  the  strength  is  nearly  in  proportion  to  the  area  of  the 
bottom  flange.  Again,  in  beams  of  this  form  which  vary  only  in 
depth,  the  strength  is  nearly  as  the  depth. 

Mr.  Hodgkinson  has  hence  deduced  the  following  simple  rule  for 
calculating  the  strength  of  cast  iron  beams  approaching  the  form  of 

greatest  strength,  viz :  W  =  — in  which  W  =  the  breaking 

weight  in  tons ;  a  =  the  area  of  bottom  flange  at  centre  of  length  in 
square  inches,  d  =  the  depth  of  the  beam  in  inches,  and  I  its  length 
in  feet. 

As  it  is  very  usual  to  express  the  load  a  girder  or  beam  has  to 
bear,  in  terms  of  its  length,  or  W=  to  w  I,  (as,  for  example,  the 
girders  of  railway  bridges  have  to  be  of  dimensions  to  bear  a  strain 
of  2  tons  per  foot  of  their  length,)  Mr.  Hodgkinson's  formula  may 
be  converted  into  the  following  very  simple  one  for  calculating  the 

area  of  the  bottom  flange,  viz  :  a  = in  which  w  is  the  weight 

2,116  d 

per  foot  of  the  girder,  of  the  load  upon  it.     Further,  as  d  is  generally 

a  simple  fraction  of  I  =  x  I,  we  may  make  the  formula  a  =  ^-^- 

25,992 

For  example,  it  is  a  usual  and  generally  convenient  proportion 
to  make  d  =  ^  I,  and  hence,  for  railway  girders,  in  which  w  =  2, 

a  =  -^t ,  which  we  may  put  a  =  — — .    Engineers  now  make 

12.99  13 

girders  of  proportions  such  as  to  bear  6  times  the  greatest  load 
likely  to  come  upon  them.  Hence,  as  there  are  generally  4  girders 
to  take  the  load  in  a  railway  bridge,  our  formula  may  be  written 

a  =  — ^t —  for  the  area  of  the  bottom  flange  (at  its  centre)  of  each 

girder. 

Open  cast  iron  girders  are  bad  in  principle.  Of  all  systems  of 
framing  girders  or  beams,  the  principle  of  perfect  continuity  of  the 
component  parts,  involved  in  Mr.  Fairbairn's  patent  malleable  iron 
girders,  is  the  best. 

Without  entering  further  into  an  examination  of  this  subject,  it 
appears  that  the  present  is  a  fitting  place  to  give  a  concise  account 
of  the  so-called  "  TUBULAR  BRIDGES,"  now  being  erected  by  Mr. 
Robert  Stephenson  for  crossing  the  Conway,  and  the  Menai  straits 
on  the  line  of  the  Chester  and  Holyhead  railway.  The  problem  of 
passing  both  these  points  with  the  "  Holyhead  road,"  was  solved  by 
Telford  in  1825,  by  the  erection  of  the  well  known  Conway  and 
Menai  suspension-bridges.  Suspension-bridges  have  been  rejected 
as  inapplicable  to  railways,  and  Mr.  Stephenson  has  proposed,  nay, 
has  already  completely  settled  the  practicability  of  carrying  out  the 
girder  system  to  meet  the  case.  A  girder  to  span  462  feet  is  an 


82 


TUBULAR  BRIDGES. 


Fig.  83. 


TUBULAR  BRIDGES.  83 

original  and  bold  conception  ;  and  now  that  it  may  be  said  to  have 
been  executed,  an  attempt,  if  only  imperfect,  to  sketch  the  progress 
of  engineering  art  in  the  direction  that  has  led  to  this  master-piece, 
cannot  but  be  useful. 

The  circumstances  demanding  or  necessitating  the  erection  of  a 
bridge  of  great  span,  occur  but  seldom,  and  the  double  condition  of 
erecting  the  bridge  without  centering,  still  more  rarely. 

The  deep  and  rapid  rivers  of  Switzerland,  seem  first  to  have  called 
forth  constructive  skill  for  this  purpose.  In  the  year  1757,  Jean 
Ulrich  Grubenmann,  born  at  Taffen,  in  the  canton  Appenzell,  erected 
the  celebrated  bridge  at  Schaffhausen,  over  the  Rhine,  in  lieu  of  a 
stone  bridge  that  had  been  swept  away  by  the  stream.  In  design- 
ing his  bridge,  Grubenmann  took  advantage  of  a  rock  about  mid- 
way across,  for  the  erection  of  a  pier  to  support  the  ends  of  two 
frames  or  compound  girders  of  carpentry,  the  one  of  170  feet,  the 
other  193  feet  clear-bearing,  or  span. 

In  1778,  Grubenmann  and  his  brother  constructed  the  Wettingen 
bridge  over  the  Limmat,  on  the  same  principle  that  had  guided 
them  so  successfully  to  the  erection  of  that  at  Schaffhausen.  This 
bridge  had  a  clear  span  of  390  feet.* 

To  Chretien  von  Michel,  an  engraver  at  Bale,  we  are  indebted 
for  the  preservation  of  a  record  of  the  details  of  construction  of  these 
two  bridges,  viz.:  "Plans,  coupes  et  Elevations  des  trois  Fonts  de 
Bois  les  plus  remarquables  de  la  Suisse,  publics  d'apres  les  dessins 
originaux,  Basle,  1803." 

Both  these  bridges  were  burnt  by  the  French  in  1799,  the  one 
having  stood  42  years,  the  other  21  years.  Over  the  one,  stones 
weighing  25  tons  each  had  passed ;  and  over  the  other  a  division  of 
the  French  army  with  its  artillery,  in  extreme  haste.  ("  Emy,  Traite* 
de  la  Charpente.")  The  points  of  construction  in  Wittingen  bridge, 
to  which  we  would  direct  especial  attention,  are : — 

1.  The  continuity  of  the  framing,  especially  in  its  vertical  plane, 
as  perfect  as  the  nature  of  the  materials  allow. 

2.  The  introduction  of  a  roof  as  an  integral  part  of  the  construct- 
ive strength  of  the  bridge,  and  of  the  disposition  of  the  greater  mass 
of  the  timber  towards  the  top  and  bottom,  while  the  intermediate 
more  slender  part,  or  rib,  is  stiffened  at  every  15  feet  by  strongly 
framed  uprights  on  the  outside  and  inside.     The  timbers  are  laid 
nearly  horizontally,  accurately  bedded  on,  and  indented  into  each 
other,  and  bolted  together  by  numerous  wrought  iron  through-bolts. 

3.  The  circumstance  that  the  two  side  frames  of  each  were  raised 
ready  framed  into  their  positions.     This  latter  is  an  inference  from 
the  fact,  that  powerful  screw-jacks  placed  on  a  scaffolding,  supported 

*  [The  single  arch  wooden  bridge,  built  by  Lewis  Wernwag,  over  the  river  Schuylkill, 
at  Fairmount,  Philadelphia,  had  a  span  of  340  feet  4  inches,  and  a  rise  of  the  arch  in 
the  centre  of  nearly  19  feet  or  above  r]5th  of  the  chord  line.  This  bridge  had  a  triple 
beam  arch  of  timber  surmounted  by  king-posts  and  truss  braces,  with  longitudinal  ties 
above,  the  whole  being  strengthened  by  screw-bolts.  See  a  figure  of  it  in  Rees'  Cyclo., 
Amer.  Edition,  vol.  34. — AM.  ED  ] 


84 


TUBULAR  BRIDGES. 


on  piles  ("  des  verins  places  sur  des  e'chafaudages  dtablis  sur  pilotes"}, 

were    used   in    raising    the 


Fig.  84. 


Fig.  85. 


bridge  at  Schaffhausen,  and 
that  the  Limmat,  near  the 
convent  of  Wettingen,  is  of 
great  depth. 

Fig.  84  is  a  section  of  the 
bridge  of  Wettingen  at  the 
ends,  and  Fig.  85  a  section 
at  the  centre.  They  suffi- 
ciently illustrate  what  we 
have  said  above  in  reference 
to  the  principle  of  continuity, 
and  the  disposition  of  the 
roof  and  timber  of  the  frames 
generally,  in  reference  to 
the  strength  of  the  bridge. 

At  the  period  when  the 
Wettingen  bridge  was  erect- 
ed by  the  Apenzell  carpen- 
ter, the  science  of  the 
strength  of  materials  had 
scarcely  begun  to  be  formed. 
Galileo's  theory,  partially 
corrected  by  the  hypothesis 
of  Hooke  and  Leibnitz,  and 
by  the  experiments  of  Ma- 
riotte  and  Buffon,  began  to 
attract  notice  ;  but  our  pre- 
sent knowledge  of  the  me- 
chanism of  the  transverse 
strain,  resulting  from  the 
later  experiments  of  Duha- 
mel,  Rondelet  and  Barlow, 
and  the  theories  founded 
upon  them  were  undevelop- 
ed. Yet  we  find  the  essen- 
tial elements  of  these  theories 
fully  recognized  in  the  con- 
struction of  the  bridges  erect- 
ed by  the  brothers  Gruben- 
mann.  Art  is  the  mother 
of  Science. 

This  was  the  largest  bridge 
ever  erected  on  Gruben- 
mann's  principle ;  but,  in 
1772,  there  was  exhibited, 
at  the  Hotel  d'Espagne,  rue 
Dauphine,  a  model  of  a 
bridge  designed  by  one  M. 


TUBULAR  BRIDGES. 


85 


Glaus,  for  Lord  Hervey.     This  was  the  model  of   a  bridge  900  feet 
span,  to  be  thrown  across  the 

Derry.     The  model  was  20  Fig.  86. 

feet  long,  or  ^  of  the  full 
size.  The  engravings  were 
executed  by  Lerouge,  and 
Fig.  86  is  taken  from  the 
plate.  It  is  a  transverse  sec- 
tion of  the  bridge  at  about 
J  of  the  span  from  the  abut- 
ment or  pier.  The  scale 
being  about  s^. 

Grubenmann's  principle  is 
adopted.  The  frames  are 
here  again  nearly  continuous. 
They  consist  of  beams  laid 
nearly  horizontal,  indented 
into  each  other,  bolted  to- 
gether by  innumerable  long 
wrought  iron  bolts,  forming 
the  side  ribs,  and  these  were 
stiffened  laterally  by  uprights.  The  floor  and  roof  are  so  framed 
with  the  trusses  or  ribs,  as  to  form  one  great  double  box,  or  hollow 
girder,  nearly  every  pound  in  the  weight  of  which  is  available  towards 
the  absolute  strength  of  the  whole. 

This  bridge  was  never  executed ;  but  we  see  in  it  a  still  more  per- 
fect adoption  of  the  plan  of  making  the  floor  and  roof  a  part  of  the 
framing,  and  also  a  recognition  of  the  fact  that  wood  has  double  the 
resistance  to  extension,  that  it  has  to  compression ;  and,  hence,  the 
timbers  of  the  upper  part  are  arranged  conformably  to  this  fact. 
This  was  clearly  recognized  by  Grubenmann,  but  not  so  perfectly 
worked  out  in  the  construction  of  his  bridges,  as  was  done  by  Glaus. 
The  introduction  of  a  roof,  as  an  integral  part  of  the  structure,  is, 
of  course,  limited  to  cases  in  which  the  span  is  such  as  necessitates 
a  depth  of  girder  of  16  to  18  feet  at  least.  The  proportion  of  the 
depth  to  the  length  of  the  bridge  of  Wettingen  is  nearly  Jj.  (For 
further  details  of  the  construction  of  the  model,  see  "  Emy,  Traite 
de  la  Charpente,  Vol.  II.  p.  398,  and  plate  134.) 

In  Great  Britain,  the  problem  of  erecting  bridges  of  wide  span  had 
scarcely  ever  been  mooted  till  about  the  beginning  of  this  century, 
when  the  joint  influence  of  the  inventions  of  her  Dudleys,  Brindleys, 
Hargreaves,  Arkwrights,  Smeatons,  Watts,  Corts,  Wyatts,  Mylnes, 
Kennies,  Telfords,  so  rapidly  developed  the  long  latent  industrial 
genius  of  the  country,  that  in  the  short  space  of  half  a  century,  from 
being  as  low  as  any,  she  became  the  first  in  the  scale  of  nations  for 
perfection  in  internal  communication,  manufacturing  skill,  and  in 
productiveness  of  the  useful  metals,  especially  iron. 

In  the  year  1800,  the  subject  of  replacing  Old  London  Bridge, 
VOL.  ii. — 8 


86  TUBULAR  BRIDGES. 

occupied  the  attention  of  nearly  every  engineer  of  eminence,  and  of 
many  men  of  acknowledged  scientific  attainments.  At  this  period, 
the  success  of  the  Wearmouth  Bridge,  designed  by  Mr.  Wilson,  in 
1793,  and  erected  in  1796,  by  Rowland  Burdon,  and  of  that  of  Buil- 
wash,  erected  by  Telford,  1796,  seems  to  have  drawn  the  attention 
of  the  most  distinguished  engineers  to  this  material,  as  that  best 
facilitating  the  execution  of  bridges  of  great  span.  The  wonderful 
progress  of  the  iron  trade  at  this  period,  also,  had  its  influence.  The 
question  of  rebuilding  London  Bridge  was  shelved  at  this  period ; 
but  Messrs.  Telford  and  Douglas  gave  in  designs  for  spanning  the 
Thames  by  a  single  arch  of  600  feet  span,  and  the  practicability  of 
the  design  was  supported  by  the  opinions  of  Playfair,  Robison,  Watt, 
Southern,  and  others.  In  1808 — 12,  Staines  Bridge  was  erected 
by  Mr.  Wilson,  that  of  Boston  by  Mr.  Rennie,  and  that  at  Bristol 
by  Mr.  Jessop.  Vauxhall  Bridge  was  commenced,  1813,  by  Rennie, 
finished  1818,  by  Mr.  Walker.  The  magnificent  Southwark  Bridge 
was  erected  1814  to  1818,  by  Messrs.  Rennie,  father  and  son. 

The  principle  of  construction  adopted  in  all  these,  was  that  of  the 
arch.  The  cast  iron  was  framed  so  as  to  render  the  structure  as 
strictly  analogous  to  that  of  an  arch  of  voussoirs  as  possible.  We 
shall  here  only  notice  that  the  adoption  of  this  principle  involves  a 
prodigious  expenditure  of  cast  iron,  to  insure  the  lateral  stability, 
essential  in  the  voussoir  principle,  beyond  what  is  necessary  for  the 
vertical  strength  required  to  bear  the  load. 

The  use  of  cast  iron  as  the  framing  of  machinery,  floor-girders, 
lock-gates,  swivel-bridges,  &c.  &c.,  became  more  and  more  usual  in 
the  construction  of  works  executed  after  1808,  at  which  period 
Brunei,  by  demonstrating  the  practicability  by  using  cast  iron  as 
the  framing  of  his  block-machinery,  gave  new  confidence  in  adopt- 
ing the  recommendation  of  Smeaton,  on  this  subject,  made  50  years 
earlier. 

In  1817,  Barlow's  Essay  on  the  "  Strength  of  Timber,  Iron,  and 
other  materials,"  was  published,  and  English  engineers  were  thus 
put  far  on  the  way  of  making  "  principles  of  science  rules  of  their 
art."  A  few  years  afterwards,  Tredgold's  Essay  "  On  the  strength 
of  cast  Iron  and  other  Metals,"  was  published;  and  this  remarkable 
work  of  a  most  remarkable  man,  together  with  Barlow's  work,  had — 
all  engineers  will  admit — a  powerful  influence  in  extending  the  ra- 
tional use  of  iron  in  construction.  Ten  years  later,  Mr.  Eaton 
Hodgkinson,  of  Manchester,  began  a  course  of  inquiry  on  the 
strength  of  iron,  which,  while  it  has  earned  for  him  and  his  coad- 
jutor, Mr.  Fairbairn,  a  high  reputation  for  scientific  knowledge  and 
skill,  has,  even  more  directly  than  the  earlier  works  mentioned,  con- 
tributed to  the  present  important  position  of  iron  as  a  material  in 
construction. 

During  this  period,  too,  the  dependence  of  England  on  Russia  and 
Sweden,  for  malleable  iron,  was  put  an  end  to,  by  the  improvements 
and  vast  extension  of  the  Welsh  and  Staffordshire  rolling-mills, 
which,  towards  1810,  began  to  stock  the  markets  with  iron,  equal 


TUBULAR  BRIDGES.  87 

for  all  ordinary  purposes  to  that  which,  up  to  this  period,  had  been 
chiefly  supplied  by  foreigners. 

It  is  a  distinguishing  element  in  the  engineer's  art,  to  adopt  the 
material  best  suited,  economically  speaking,  to  the  work  he  has  to 
accomplish. 

In  1806,  the  price  of  bar  iron,  larger  size,  was  £20  per  ton ;  in 
1816,  it  was  £10  per  ton ;  in  1828,  it  was  £8  per  ton ;  and  in  1831, 
it  was  £5  to  £6  per  ton. 

Thus  this  material  has  gradually  come  into  the  domain  of  applica- 
tions in  construction,  from  which  its  high  price  had  long  excluded 
the  consideration  of  its  qualifications.  Roofs  of  great  span  began 
to  be  formed  of  combinations  of  cast  and  malleable  iron.  The  eligi- 
bility of  the  one  to  resist  strains  of  compression,  and  of  the  other  to 
resist  tensile  strains,  became  familiar  to  those  engaged  in  practical 
construction. 

In  1825,  a  new  engineering  era  had  arisen.  As  the  genius  of 
Brindley,  under  the  mighty  influence  of  the  policy  of  a  Chatham, 
had  created  the  inland  navigation  of  England,  the  genius  of  a  Ste- 
phenson,  under  the  influence  of  the  policy  of  a  Huskisson,  created 
the  railway  system.  Steam  navigation  advanced  from  mere  essays 
to  a  system  of  vast  importance.  The  demands  of  the  ship  builder, 
the  locomotive  maker,  the  railway  engineer,  gave  rise  to  new  exertions 
of  the  iron  masters.  Blooms  were  puddled,  of  sizes  hitherto  deemed 
impracticable.  It  became  usual  to  have  bars  rolled,  and  pieces  forged 
of  sizes  exceeding  those  which,  within  a  few  years,  had  been  deemed 
wonderful  or  isolated  examples.  In  this  respect,  the  complacent 
dictum  of  a  celebrated  engineer,  that  "no  difficulty  can  arise  in 
engineering  or  mechanical  art,  that  is  not  certain  to  be  overcome," 
has  been  fully  borne  out. 

In  the  construction  of  the  London  and  Birmingham  Railway,  the 
Great  Western  Railway,  the  Midland  Counties  Railway,  and  others, 
the  engineers  made  ample  use  of  cast  iron,  and  examples  of  girders 
of  50,  60,  even  70  feet  in  length  are  to  be  found  on  these  lines  of 
railway.  The  scientific  principles  of  construction  of  such  girders 
were  not  at  once  recognized  or  learned,  and  we  consequently  find 
excess  of  iron  in  most  instances,  and  mistaken  construction  in  others. 
There  was  no  time  for  gathering  exact  knowledge,  though  extant. 
A  limited  experience  of  successful  cases  led  to  endless  repetitions  of 
girders  of  not  very  happy  proportions,  and  "trussed"  in  the  wrong 
direction.  The  outcry  made  in  England  on  the  subject  of  hot  blast 
iron  being  so  inferior  in  quality,  so  treacherous,  &c.  &c.,  the  conse- 
quent high  price  demanded  for  castings  of  what  was  termed  good 
iron,  had  considerable  influence  in  limiting  the  applications  of  iron 
in  railway  bridges.  Stone  and  brick  were  preferred  for  the  few 
bridges  of  great  span  erected.  Suspension  bridges  were  tried  and 
failed.  Of  the  wooden  bridges  erected,  that  over  the  Tyne  at  Scots- 
wood,  by  Mr.  Blackmore,  deserves  mention  as  involving  the  best 
principles  of  construction.  The  path  so  well  opened  up  by  Gruben- 
mann  had  long  been  lost.  The  system  of  the  Bavarian  engineer, 


88  TUBULAR  BRIDGES. 

Wiebecking,  and  applied  by  him  successfully  to  the  bridge  at  Bam- 
bery,  215  feet  span,  and  others,  were  extensively  made  known  by 
his  published  writings,  whilst  the  better  principle  of  Grubenmann 
was  overlooked.  The  essential  part  of  Wiebecking's  system  consists 
in  putting  the  main  strength  of  the  frame  in  arches  of  curved  timbers 
trenailed  together,  on  to  which  the  rest  of  the  timbers  of  each  truss 
is  framed,  suspending  the  horizontal  ties,  from  which  the  road-way 
is  supported.  Wiebecking's  system,  with  certain  modifications,  was 
adopted  in  France  by  M.  Emmery,  about  1830,  and  by  the  Messrs. 
Green,  of  Newcastle,  about  1840.  In  imitation  of  Wiebecking's  plan, 
too,  the  bow  and  string  fashion  of  open  cast  iron  girders  was  adopt- 
ed, small  as  is  the  analogy  between  wood  and  iron.  Beginning  with 
the  bridge  over  the  Regent's  Canal  at  Camden  Town,  this  fashion 
of  girder  has  been  many  times  repeated,  on  various  scales ;  and  is  in 
execution  even  at  the  present  moment,  for  spans  of  120  feet,  in  the 
high  level  bridge  at  Newcastle-upon-Tyne. 

In  the  mean  time,  in  America,  Town's  lattice  frame  bridges,  and 
Long's  diagonal  frame  bridges,  had  been  invented,  and  railway 
bridges  of  150  to  180  feet  clean  span,  had  been  executed  accord- 
ing to  each  system.  In  the  largest  application  of  Long's  system, 
the  depth  of  the  frame  is  about  20  feet,  and  the  sides  and  floor,  and 
roof  are  connected  together,  so  as  to  form  one  box-like  girder.  The 
diagonal  framing,  even  when  carried  out  in  the  form  of  lattice  work, 
makes  but  an  imperfect  continuity  in  the  framing,  or  ribs  connecting 
together  the  top  and  bottom  rails  or  flanges ;  but  this  is  the  principle 
aimed  at,  and  the  bridges  are  to  be  considered  as  very  successful 
engineering.  They  have  been  adopted  in  England,  in  a  few  cases, 
the  largest  being  that  of  an  occupation  bridge  on  the  Birmingham 
and  Gloucester  railway ;  but  wooden  structures  are  avoided  in  that 
country,  on  account  of  the  extreme  variations  in  the  hygrometric 
state  of  the  atmosphere. 

Of  the  many  lattice  bridges  erected  in  America,  the  most  interest- 
ing in  reference  to  our  subject,  is  the  iron  tubular  lattice  bridge  in 
the  great  hotel,  Tremont  House,  at  Boston.  This  is  an  elliptical 
tube  of  lattice  or  trellis  work,  the  height  being  7  to  8  feet,  the  minor 
axis  of  the  ellipse  being  4'— 6",  the  span  about  120  feet.  The  top 
is  stiffened  by  a  longitudinal  bar.  The  flooring  of  wood  on  the  bot- 
tom, is  about  three  feet  6  inches  wide,  and  helps  to  stiffen  the  whole. 
This  foot  bridge  had  been  several  years  in  use  in  1843,  and  its  per- 
fect rigidity,  it  may  be  here  mentioned,  at  once  suggested  the  appli- 
cability of  the  plan  for  carrying  a  railway  across  the  Menai  straits. 

Among  the  circumstances  concurring  to  the  result  consummated 
by  Mr.  Stephenson,  the  success  of  iron  ships  of  enormous  dimen- 
sions, in  resisting  the  strain  they  have  to  undergo,  is  certainly  a 
prominent  one.  The  Great  Britain  steam-ship,  for  example,  is  253 
feet  in  length.  It  is  mainly  composed  of  sheet  and  angle  iron,  of 
less  than  half  an  inch  in  thickness;  it  is  thus,  like  other  iron  ships, 
a  mere  shell ;  and  yet  from  its  perfect  continuity,  and  the  nature  of 


TUBULAR  BRIDGES.  89 

the  materials,  has,  unimpaired,  withstood  lateral  strains  under  which 
a  vessel,  on  almost  any  other  construction,  must  have  broken  up. 

Such  was  the  state  of  preparation  of  engineers'  minds  for  solving 
the  problem  of  carrying  a  railway  across  the  Menai  straits  by  girders, 
when,  early  in  1845,  Mr.  Stephenson's  "aerial  tunnel"  was  spoken 
of.  On  the  5th  of  May,  1845,  he  announced  his  plan  before  a  com- 
mittee of  the  House  of  Commons.  * 

Few  inventors  can  explain  the  development  in  their  minds  of  an 
original  conception.  Invention  in  art  consists  of  two  distinct  intel- 
lectual efforts — first,  in  seizing  the  ideal  conception  of  the  object  to 
be  made  for  a  given  end ;  and  second,  in  the  contrivance  of  the  suit- 
able arrangement  of  materials  (or  of  mechanism,  in  the  case  of  a 
machine)  for  that  object.  The  nature  of  the  first  conception  seems 
always  to  depend  on  the  existing  state  of  analogous  objects,  and, 
hence,  the  two  parts  of  the  process  are  generally  intimately  con- 
nected, though  not  inseparable.  In  Mr.  Stephenson's  case,  the  two 
processes  seem  to  have  been  separated.  For  as  early  as  April,  1845, 
Mr.  Eaton  Hodgkinson  and  Mr.  Fairbairn  seem  to  have  been  con- 
sulted as  to  experiments  on  the  strength  of  cylindrical  tubes  of 
riveted  sheets  of  iron,  and  as  to  the  necessity  of  a  combination  of 
the  girder  plan  with  suspension  chains,  for  his  great  bridges.  We 
learn  from  a  communication  of  Mr.  Hodgkinson's  to  the  Mechanical 
Section  of  the  meeting  of  the  British  Association,  held  at  Southamp- 
ton, in  1846,  "  that  a  number  of  experiments  were  made  upon  cylin- 
drical and  elliptical  tubes,  and  a  few  upon  rectangular  ones ;"  but, 
inasmuch  as  a  girder  has  to  resist  in  its  vertical  direction  much  more 
than  in  its  horizontal,  the  oblong  rectangular  form  should  have 
immediately  suggested  itself  as  the  best ;  and,  therefore,  these  first 
experiments  were  works  of  supererogation. 

Mr.  Hodgkinson's  experiments  were,  therefore,  at  once  directed  to 
ascertaining  what  should  be  the  distribution  of  the  metal  in  hollow 
rectangular  girders,  to  secure  a  maximum  of  strength  with  a  minimum 
of  weight.  Mr.  Hodgkinson,  whose  investigations,  published  in  1840, 
had  proved  experimentally  that  hollow  columns  have  a  greater  re- 
sistance to  compression  than  the  same  weight  of  material  in  a  solid 
column  (as  the  usual  theory  had  indicated,  and  the  practice  of 
Wiebecking  and  Gauthey  thirty  years  earlier,  and  of  Polonceau,  in 
1839,  had  testified),  now  made  further  experiments  to  ascertain  the 
relative  resistance  of  circular  and  rectangular  tubes,  with  the  object 
of  disposing  of  the  malleable  iron,  of  which  the  girders  were  to  be 
made  in  this  hollow  form,  on  the  upper  side,  i.  e.,  the  part  com- 
pressed by  the  strain. 

As  might  have  been  anticipated,  the  "  buckling"  of  the  plates  on 
the  top  had  to  be  prevented  by  particular  contrivances,  or  by  greatly 
increasing  their  substance  beyond  that  of  the  bottom  or  extended 
side. 

The  following  are  some  of  the  leading  results  of  Mr.  Hodgkinson's 
experiments. 

Experiments  on  two  similar  tubes. 
8* 


90 


TUBULAR  BRIDGES. 


Length 
of 
tube. 

Weight 
of 
tube. 

Distance 
between 
supports. 

Depth 
of 
tube. 

Breadth 
of 
tube, 

Thickness  of 
metal  in  16ths 
of  an  inch. 

Breaking 
weight  in 
tons. 

Ultimate 
deflexion. 

31'—  6" 

cwt.  qu. 
20  —  3 

feet 
2 

l'_  4" 

Top  Bottom  Side 
642 

26,1 

Inches 
2} 

47—0 

61—1 

45 

3 

2'—  0 

963 

65,5 

3 

This  breaking  weight  in  tons  is  in  excess  of  the  results  deduced 
from  the  usual  formula,  when  the  value  of  /  (the  moment  of  inertia), 
is  calculated  by  our  formula  5  (page  79),  when/1  is  taken  =  56000. 
To  ascertain  the  power  of  such  tubes  to  resist  a  lateral  strain — as 
from  the  action  of  wind — the  smaller  of  these  two  tubes,  after  being 
well  repaired,  was  laid  on  its  side  and  broken.  The  mean  of  two 
experiments  gave  15,2  tons  as  breaking  weight,  which  is  about  25 
per  cent,  above  the  result  of  calculation  by  our  formulas,  when  the 
value  of  y1  is  taken  as  indicated.  Experiments  on  the  strength  of 
sheet  iron,  however,  give  the  tensile  resistance  as  high  as  62000 
Ibs.  per  square  inch,  and  if  we  introduce  this  as  the  value  of  /',  the 
experimental  results  would  almost  exactly  correspond  with  the  re- 
ceived theory. 

Mr.  Hodgkinson's  experiments  on  the  resistance  of  sheet  iron  tubes 
to  compression,  show  (as  his  experiments  on  cast  iron  columns  made 
in  1839,  had  previously  done,  and  as  Euler's  theory  indicates),  that 
rectangular  tubes  are  weaker  than  square  ones,  and  both  of  these 
much  weaker  than  cylindrical  tubes ;  so  much  so,  indeed,  that  the 
substitution  of  cylindrical  for  square  or  rectangular  tubes,  would, 
according  to  Mr.  Hodgkinson's  experiments,  effect  a  saving  of  one- 
fourth  of  the  metal  in  the  top. 

Mr.  Fairbairn,  at  the  same  meeting  of  the  British  Association, 
September,  1846,  made  the  following  communication  of  "Experi- 
ments on  the  Tubular  Bridge,  proposed  by  Mr.  R.  Stephenson,  for 
crossing  the  Menai  straits.  These  experiments,  says  Mr.  Fair- 
bairn,  have  put  us  in  possession  of  facts,  which  greatly  increase  our 
knowledge  of  the  properties  of  a  material,  whose  powers,  when  it  is 
properly  put  together,  are  but  imperfectly  understood ;  for  exclusive 
of  the  rapidly  increasing  use  of  wrought  iron  in  the  construction  of 
ship-boilers,  £c.,  its  application  to  bridges  of  the  tubular  form  is 
perfectly  novel,  and  originated  with  Mr.  Robert  Stephenson.  Ex- 
periments of  the  most  conclusive  character  were  those  made  on  a 
model  tube  on  a  large  scale,  containing  nearly  all  the  elements  of 
the  proposed  bridge,  and  the  various  conditions  with  regard  to  form 
and  construction,  which  had  been  developed  by  the  previous  inquiries 
(above  alluded  to).  It  occurred  to  Mr.  Fairbairn  that  the  strongest 
form  would  be  that,  wherein  the  top  and  bottom  consisted  of  a  series 
of  pipes,  with  riveted  plates  on  their  upper  and  under  sides.  This 
form  of  top,  says  Mr.  Fairbairn,  would  possess  great  rigidity,  and  is 
well  adapted  to  resist  the  crushing  forces  to  which  it  is  subjected ; 
and  the  bottom  section  appeared  equally  powerful  to  resist  tension. 


TUBULAR  BRIDGES. 


91 


Fig.  87. 


Mr.  Fairbairn  thought  that  this  is  the  strongest  form  that  could  be 
devised ;  but  practical  difficulties  present  themselves  in  its  construc- 
tion, as  an  easy  access  to  the  different  parts  for  the  purposes  of 
painting,  repairs,  &c.,  is  absolutely  necessary.  The  scale  of  the 
model  tube  was  exactly  one-sixth  of  the  length,  breadth,  depth,  and 
thickness  of  metal  of  the  bridge  intended  to  cross  one  span  of  the 
straits,  450  feet,  (since  increased  to  462  feet.)  In  each  of  the  expe- 
riments, the  weights  were  laid  on  at  the  centre,  about  one  ton  at  a 
time,  and  the  deflection  was  carefully  taken  as  well  as  the  defects  of 
elasticity  after  the  load  was  removed. 

"The  rectangular  model  tube,  Fig.  87,  was  80  feet  long,  4'— 6" 
deep,  2' — 8"  wide,  75  feet  between  the 
supports.  The  thickness  of  the  plate: 
bottom  .156  inch,  sides  .099  inch,  top 
.147  inch,  sectional  area  of  bottom  8,8 
inches,  weight  of  the  tube  4,86  tons  = 
10,889  Ibs.  First  experiment,  breaking 
weight  79,578  Ibs  =35f  tons.  Ultimate 
deflexion  4,375  inches,  permanent  set  un- 
der strain  of  67,842  Ibs.  .792  inch.  With 
the  strain  of  35  J  tons,  the  bottom  was  torn 
asunder,  directly  across  the  solid  plates, 
at  a  distance  of  2  feet  from  the  centre  of 
the  shackle,  from  which  the  load  was  sus- 
pended. One  of  the  principal  objects  of 
this  inquiry  was  to  determine  the  ratio 
between  the  top  and  bottom  of  the  tube. 
From  the  experiments  immediately  preceding  this,  it  appeared  that 
the  ratio  of  the  area  of  the  top  to  that  of  the  bottom,  in  a  rectangu- 
lar tube  (of  thin  sheet  iron),  should  be  as  5  to  3. 

"  The  plates  forming  the  top  of  the  model  tube  were  somewhat 
thicker  than  intended,  and  consequently  gave  (as  former  experiments 
indicated)  a  preponderating  resistance  to  that  part.  To  obviate  this 
disparity,  two  additional  strips,  6J  by  Y5g,  weighing  about  4  cwt. 
were  riveted  along  the  bottom,  extending  20  feet  on  each  side  the 
centre.  This  raised  the  area  of  the  bottom  to  nearly  13  inches, 
being  about  the  ratio  of  5  to  3,  or  23,5  to  13.  With  these  propor- 
tions, and  having  repaired  the  fractured  part  by  introducing  new 
plates,  the  experiments  proceeded  as  before. 

"Second  experiment.  Breaking  weight  97,102  Ibs.  =  43,3  tons. 
Ultimate  deflexion  4,11  inches.  In  this  experiment  the  tube  failed 
by  one  of  the  ends  giving  way,  which  caused  the  sides  to  collapse. 
The  weak  point  in  this  girder  was  evidently  a  want  of  stiffness  in 
the  sides.  To  remedy  this  evil  and  keep  them  in  form,  vertical  ribs, 
composed  of  light  angle  iron,  were  riveted  along  the  interior  of  each 
side  at  distances  of  2  feet;  and,  having  again  restored  the  injured 
parts,  the  tube  was  a  third  time  subjected  to  the  usual  tests. 

"Third  experiment.  Breaking  weight  126,138  Ibs.  =56,3  tons, 
ultimate  deflexion  5,68  inches.  The  tube  was  torn  asunder  through 


92 


TUBULAR  BRIDGES. 


the  bottom  plates.  The  cellular  top  gave  evident  symptoms  of  yield- 
ing to  a  crushing  forge  by  the  puckerings  of  each  side,  which  gra- 
dually enlarged  as  the  deflection  increased.  These  appearances 
became  more  apparent  as  the  joints  of  the  plates  on  the  top  side  had 
sheared  off  a  number  of  the  rivets,  and  the  holes  had  slid  over  each 
other  to  an  extent  of  nearly  T3a  of  an  inch." 

On  Mr.  Fairbairn's  most  admirably  stated  facts,  we  shall  only 
remark,  that  a  cellular  bottom  would  probably  be  found  to  be  the 
weakest  and  not  the  strongest  form  in  which  the  iron  could  be  dis- 
tributed there;  for  there  is  no  tendency  to  buckle  in  the  bottom;  and 
to  resist  the  transverse  strain  of  passing  loads  (in  the  actual  bridge), 
the  separation  of  the  plates  composing  the  bottom,  should  only  be 
such  as  to  allow  of  the  introduction  of  connecting  plates  or  joists  to 
stiffen  it,  so  as  to  make  the  bottom  a  roadway.  Again,  the  ratio  of 
the  areas  of  the  top  and  bottom  above  deduced,  is  evidently  not  an 
absolute  quantity,  but  refers  only  to  the  particular  form  of  cells 
adopted  in  these  experiments.  Theory  and  experiment  indicate  this 
to  be  the  true  view  of  the  case. 

These  experiments  were  used  in  determining  the  dimensions  of  the 

bridges  already  erected, 


Fig.  88. 


and  now  in  construction. 
In  reference  to  the  Con- 
way  Bridge,  the  first 
tube  of  which  was  erect- 
ed in  March,  1848,  the 
following  particulars  are 
taken  from  the  Civil  En- 
gineers' and  Architects' 
Journal,  for  June,  of  this 
year. 

"Fig.  88  exhibits  a 
transverse  section  of  one 
of  the  tubes.  Fig.  89  is 
a  side  elevation,  of  12 
feet  in  length,  of  the  tube, 
resting  on  the  masonry. 
The  tube  consists  of  sides 
a,  a,  of  wrought  iron 
plates,  from  4  to  8  feet 
long,  and  2  feet  wide,  by 
\  inch  thick  in  the  cen- 
tre, and  f  ths  of  an  inch 
thick  towards  the  end 
of  the  tube,  riveted  to- 
gether to  T -angle-iron 
ribs,  placed  on  both  sides 
of  the  joints,  and  angle- 
gussets  at  the  feet  of  the  ribs  to  stiffen  them ;  a  ceiling  (or  top 
lange),  composed  of  8  cells  or  tubes  6,  each  20£  inches  wide,  and 


TUBULAR  BRIDGES. 


Fig.  89. 


21  inches  high;  and  a  floor  containing  6  cells  or  tubes  c,  27 J  inches 
wide,  and  21  inches  high.  The  whole  length  of  the  tube  is  412  feet; 
it  is  22  feet  3|  inches  high 
at  the  ends,  and  25  feet  6 
inches  high  in  the  centre, 
(including  the  tubes  at  top 
and  bottom,  running  the 
whole  length,)  and  14  feet 
wide  to  the  outside  of  the 
side  plates.  The  upper  cells 
are  formed  of  wrought  iron 
plates,  f  inch  thick  in  the 
middle,  and  J  inch  thick 
towards  the  ends  of  the 
tube,  put  together  with  an- 
gle-iron in  each  angle  of  the 
cells;  and  over  the  upper 
joints  is  riveted  a  slip  of  J 
inch  iron,  4|  inches  wide. 
The  lower  cells  consist  of 
|  inch  iron  plates  for  the 
divisions,  and  the  top  and 
bottom  of  two  thicknesses 
of  plate,  each  12  feet  long, 
2  feet  4  inches  broad,  and 
J  inch  thick  in  the  centre, 
and  £  inch  thick  at  the  ends, 
and  so  arranged  as  to  break 
the  joint;  and  a  covering  plate  of  \  inch  iron,  3  feet  long,  is  placed 
over  every  joint  on  the  underside  of  the  tube.  The  external  casing 
is  united  to  the  top  and  bottom  cells  by  angle-iron,  on  both  the 
inside  and  outside  of  the  tube.  The  ends  of  the  tube,  where  it  rests 
on  the  masonry,  are  strengthened  by  cast  iron  frames  d,  to  the 
extent  of  8  feet  of  the  lower  cells.  The  tube  was  constructed  on  a 
platform  erected  on  the  shore  of  the  river,  close  to  where  it  was  to 
cross;  and,  when  finished,  six  pontoons  were  placed  under  the  tube 
at  low  water,  and  at  high  water  they  lifted  the  tube  off  the  piles 
upon  which  the  stage  was  erected.  It  was  then  floated  to  its  desti- 
nation, and  placed  between  the  two  towers,  part  of  the  masonry  being 
left  undone  until  the  tube  was  put  into  its  proper  position,  and  as  it 
was  raised  by  means  of  hydraulic  lifting  presses,  the  masonry  was 
built  up  under  the  tube.  In  order  to  allow  of  the  free  expansion 
and  contraction  of  the  tube,  the  ends  rest  on  24  pairs  of  iron  rollers 
z,  connected  together  by  a  wrought  iron  frame,  and  placed  between 
two  cast  iron  plates  j,  k,  12  feet  long  by  6  feet  wide,  and  4  inches 
thick.  The  lower  plate  is  laid  on  a  flooring  of  3  inch  planks  I,  bedded 
on  the  stone-work.  Fig.  88,  A,  A,  are  uprights,  into  which  are  fitted 
the  cross-lifting  girders  for  attaching  the  chains  of  the  hydraulic 
presses." 


94  TUBULAR  BRIDGES. 

The  weight  of  each  tuhe  of  the  Con  way  Bridge  has  been  stated 
to  be  1300  tons,  but  whether  this  is  the  weight  including  the  fixtures 
for  the  rails,  or  of  the  tube  per  se,  is  not  recorded  in  the  papers  to 
which  we  have  had  access.  The  total  length  being  420  feet,  the 
weight  may  be  stated  as  not  less  that  62  cwt.  or  3,1  tons  per  "  foot 
running."  It  is  difficult  to  conceive  anything  more  admirable  than 
this  final  result,  when  we  learn  that,  under  the  passage  of  the  heaviest 
goods-trains,  there  is  no  sensible  motion,  by  deflection,  among  the 
parts  of  the  tube-girders.  No  method  of  construction,  hitherto 
adopted  for  large  spans,  could  have  accomplished  this  absolute  secu- 
rity with  so  small  a  weight  of  materials. 

In  what  precedes,  we  have  endeavored  to  trace  the  progress 
of  a  particular  part  of  the  engineer's  art,  with  a  view  to  encourage 
young  engineers  to  look  upon  their  art  as  capable  of  being  formed 
into  a  science ;  and  we  yet  venture  to  add,  in  reference  to  what  we 
have  said  as  to  the  separate  intellectual  efforts  involved  in  Mr.  Ste- 
phenson's  invention — that  the  adoption  of  the  results  of  the  above- 
mentioned  experiments,  the  execution  of  the  designs  ultimately 
determined  upon,  and  the  erection  of  the  tubes,  are  details  requiring 
the  highest  order  of  skill  and  practice  in  the  execution  of  works ; 
but,  the  "  keeping  hold  of  the  original  idea,"  until  brought  to  the 
form  of  ascertaining,  experimentally,  the  best  shape  or  arrangement 
of  the  materials,  is  certainly  the  essence  of  invention — marks  out  the 
Engineer-in-chief 's  work  unmistakably — is  the  element  in  the  grand 
result  that  commands  the  homage  paid  to  engineering  genius,  by  the 
less  gifted  of  the  profession,  and  secures  to  itself,  envy  or  jealousy 
notwithstanding,  the  due  meed  of  fame  and  public  applause.  The 
co-operation  of  Mr.  Hodgkinson,  Mr.  Fairbairn,  and  Mr.  Clark, 
has  doubtless  been  of  very  great  service  to  Mr.  Stephenson,  and 
the  Holyhead  Railway  Company,  in  working  out  the  details  of  a 
design,  in  proposing  which,  however,  they  had  no  share ;  but  it 
seems  impossible  to  associate  the  names  of  others  with  that  of  Ste- 
phenson in  this  work,  further  than  we  associate  the  names  of  Davies, 
Gilbert,  Rhodes,  and  Provis,  with  that  of  Telford,  in  connection 
with  the  Menai  Suspension  Bridge.  Just  as  well  might  "  all  but 
the  original  idea"  of  the  block  machinery  be  claimed  by  Maudslay, 
who  made  it  for  Brunei.  The  same  of  the  safety-lamp,  by  the  tin- 
smith, who  made  the  first  for  George  Stephenson.  The  same  of  the 
hot-blast,  by  Mr.  Wilson  or  Mr.  Condie,  who  first  applied  it  for  Mr. 
Neilson ;  and  the  same  of  many  other  cases,  in  which,  from  impera- 
tive circumstances,  the  inventor  has  found  himself  necessitated  to 
delegate  to  others  the  actual  execution  of  his  "  original  idea." 

Whilst  the  experiments  on  the  Tubular  Bridges  were  in  progress, 
letters-patent  were  granted  to  Mr.  Fairbairn,  for  "  improvements  in 
constructing  iron  beams;"  in  which  he  claims  "the  novel  application 
and  use  of  plates  or  sheets  of  iron,  united  by  means  of  angle  iron 
and  rivets,  or  by  other  means,  for  forming  or  constructing,  by  such 
combination,  hollow  beams  or  girders  for  the  erection  of  bridges  or 
other  buildings." 


STRENGTH  OF  STEAM  BOILERS,  TUBES,  ETC.  95 

Although  the  principle  of  Mr.  Fairbairn's  patent  is  perfect,  the 
limits  of  its  economical  application,  is,  as  far  as  we  can  judge,  to 
cases  of  great  span,  and  where  extreme  strains  are  likely  to  be  sud- 
denly brought  upon  the  structure. 

The  system  successfully  applied  by  Wiebecking,  for  wooden 
bridges,  and  in  cast  iron,  by  Reichenbach  and  Gauthey,  and,  lat- 
terly, to  a  certain  extent,  by  Polonceau,  in  the  beautiful  Pont  du 
Carrousel,  over  the  Seine,  has  recently  been  revived  in  malleable 
iron.  On  the  extension  of  the  London  and  Blackwall  Railway,  and 
elsewhere,  Mr.  Locke  has  introduced  a  hollow  sheet  iron  arch,  with 
suspended  tie  and  diagonal  bracings,  to  carry  the  roadway.  We 
object  to  this  form  of  bridge,  as  being  a  retrogression  in  the  principle 
of  construction,  and  consequently  offering  no  economical  advantage, 
but  the  contrary.* 

[STRENGTH  OF  CYLINDRICAL  STEAM  BOILERS,  TUBES,  AND  FIRE-ARMS. 

It  has  been  generally  supposed  that  the  rolling  of  boiler-plate  iron, 
gives  to  the  sheets  a  greater  tenacity  in  the  direction  of  the  length, 
than  in  that  of  the  breadth.  Supposing  this  to  be  correct,  it  has 
frequently  been  asked  how  the  sheets  ought  to  be  disposed  in  a 
cylindrical  boiler  of  the  common  form,  in  order  to  oppose  the  greatest 
strength  to  the  greatest  strain.  It  has  also  been  asked  whether  the 
same  arrangement  will  be  required  for  all  diameters,  or  whether  a 
magnitude  will  not  be  eventually  attained,  which  may  require  the 
direction  of  the  sheets  to  be  reversed? 

To  determine  these  questions  in  a  general  manner  recourse  must 
be  had  to  mathematical  formulas,  assuming  such  symbols  for  each 
of  the  elements  as  may  apply  to  any  given  case  of  which  the  sepa- 
rate data  are  determined  either  by  experiment  or  by  the  conditions 
of  the  case.  The  principles  of  the  calculation  require  our  first  notice. 

1.  To  know  the  force  which  tends  to  burst  a  cylindrical  vessel  in 
the  longitudinal  direction,  or,  in  other  words,  to  separate  the  head 
from  the  curved  sides,  we  have  only  to  consider  the  actual  area  of 
the  head,  and  to  multiply  the  number  of  units  of  surface  by  the 
number  of  units  of  force  applied  to  each  superficial  unit.     This  will 
give  the  total  divellent  force  in  that  direction. 

To  counteract  this,  we  have,  or  may  be  conceived  to  have,  the 
tenacity  of  as  many  longitudinal  bars  as  there  are  lineal  units  in  the 
circumference  of  the  cylinder.  The  united  strength  of  these  bars 
constitutes  the  total  retaining  or  quiescent  force;  and,  at  the  moment 
when  rupture  is  about  to  take  place,  the  divellent  and  the  quiescent 
forces  must  obviously  be  equal. 

2.  To  ascertain  the  amount  of  force  which  tends  to  rupture  the 


*  Of  all  the  designs  for  iron  bridges  hitherto  planned  and  executed — when  we  con- 
sider the  situation,  the  extent,  the  elevation  above  the  water,  and  the  scenery  by  which 
it  is  surrounded — the  most  imposing  is  that  of  the  wire  bridge  over  Niagara  river,  a  short 
distance  below  the  Falls.  This  bridge  is  still  in  progress.  It  was  planned  and  executed 
by  Mr.  Charles  Ellet,  Jr.— AM.  ED. 


96  STRENGTH  OF  STEAM  BOILERS,  TUBES,  ETC. 

cylinder  along  the  curved  side,  or  rather  along  two  opposite  sides, 
we  may  regard  the  pressure  as  applied  through  the  whole  breadth 
of  the  cylinder  upon  each  lineal  unit  of  the  diameter.  Hence  the 
total  amount  of  force  which  would  tend  to  divide  the  cylinder  in 
halves  by  separating  it  along  two  lines,  on  opposite  sides,  would  be 
represented  by  multiplying  the  diameter  by  the  force  exerted  on 
each  unit  of  surface,  and  this  product  by  the  length  of  the  cylinder. 
But  even  without  regarding  the  length,  we  may  consider  the  force 
requisite  to  rupture  a  single  band  in  the  direction  now  supposed, 
and  of  one  lineal  unit  in  breadth ;  since  it  obviously  makes  no  differ- 
ence whether  the  cylinder  be  long  or  short  in  respect  to  the  ease  or 
difficulty  of  separating  the  sides.  The  divellent  force  in  this  direc- 
tion is,  therefore,  truly  represented  by  the  diameter  multiplied  by 
the  pressure  per  unit  of  surface.  The  retaining,  or  quiescent  force, 
in  the  same  direction,  is  only  the  strength  or  tenacity  of  the  two 
opposite  sides  of  the  supposed  band.  Here,  also,  at  the  moment 
when  a  rupture  is  about  to  occur,  the  divellent  must  exactly  equal 
the  quiescent  force. 

3.  In  order  to  estimate  the  augmentation  of  divellent  force  conse- 
quent upon  an  increase  of  diameter,  we  have  only  to  consider,  that, 
as  the  diameter  is  increased,  the  product  of  the  diameter,  and  the 
force  per  unit  of  surface,  is  increased  in  the  same  ratio.     But  unless 
the  thickness  of  the  metal  be  increased,  the  quiescent  force  must 
remain  unaltered.      The  quiescent  forces,  therefore,   continue  the 
same — the  divellent  increase  with  the  diameter. 

4.  Again,  as  the  diameter  of  the  cylinder  is  increased,  the  area 
of  its  end  is  increased  in  the  ratio  of  the  square  of  the  diameter. 
The  divellent  force  is,  therefore,  augmented  in  this  ratio.     But  the 
retaining  force  does  not,  as  in  the  other  direction,  remain  the  same, 
since  the  circumference  of  a  circle  increases  in  the  same  ratio  as  the 
diameter.     The  quiescent  force  will,  consequently,  be  augmented  in 
the  simple  ratio  of  the  diameter,  without  any  additional  thickness 
of  metal.     So  that,  on  the  whole,  the  total  tendency  to  rupture  in 
this  direction  will  increase  only  in  the  simple  ratio  of  the  diameter. 

5.  Since  we  have  seen  that  the  tendency  to  rupture,  in  both  direc- 
tions, increases  in  the  simple  direct  ratio  of  the  increase  of  diameter, 
it  is  obvious  that  any  position  of  the  sheets  which  is  right  for  one 
diameter,  must  be  right  for  all.     Hence  there  can  never  be  a  condi- 
tion, in  regard  to  mere  magnitude,  which  will  require  the  sheets  to 
be  reversed. 

6.  The  foregoing  considerations  being  once  admitted,  we  may 
proceed  to  ascertain  what  is  the  true  direction  of  the  greatest  tena- 
city in  the  sheet,  if  any  difference  exist,  and  what  that  difference 
might  amount  to,  consistently  with  equal  safety  of  the  boiler  in  both 
directions. 

7.  Let  x=  the  diameter  of  the  cylinder. 

f=  the  force  or  pressure  per  unit  of  surface  (pounds  per  square 
inch,  for  example). 

T  =  the  tenacity  of  metal  which,  with  the  diameter  z,  and  the 


STRENGTH  OF  STEAM  BOILERS,  TUBES,  ETC.  97 

force  y,  will  be  required  in  the  lineal  unit  of  the  circumference,  in 
order  to  hold  on  the  head. 

Then  will  the  whole  quiescent  force  be  3. 1416  a;  T,  while  the  divel- 
lent  will  be  .7854  x2f;  consequently,  .7854  or2/ =  3. 1416 a;  as  above 
stated. 

Dividing  by  . 7854  x,  we  have  xf=  4T  ;  and  we  derive  immedi- 
ately— 

4T 

X=^ 


and  T  =       . 
4 

That  is,  the  tenacity  of  the  longitudinal  bar  of  the  assumed  unit  in 
width,  will  be  one-fourth  of  the  product  of  the  diameter  into  the  pres- 
sure, measuring  the  tenacity  by  the  same  standard  as  the  pressure, 
whether  in  pounds  or  kilograms. 

8.  Now  assuming  the  tenacity  required  in  the  circular  band  of  the 
same  width  to  be  t,  we  shall,  agreeably  to  what  has  already  been 
snid,  have  the  divellant  force  expressed  by  xf,  and  the  quiescent  by 

2t,  so  that  */*=  2t  and  t  =  *£-.     Also/=  —',  and  x  =  —. 

Having  thus  obtained  two  expressions  for  each  of  the  quantities  x 
and  f,  we  may,  by  comparing  them,  readily  discover  the  relative 
values  of  T  and  t, — thus  : 


« 


4T       2f 
}.  hence  _  =  _  whence  4T  =  2*,  or  t  =  2T.    From  which 

^T         /    / 

it  follows,  that  under  a  known  diameter,  and  with  a  given  force  or 
pressure,  the  tenacity  of  metal  in  a  cylindrical  boiler  of  uniform  thick' 
ness,  ought  to  be  twice  as  great  in  the  direction  of  the  curve  as  in  that 
of  the  length  of  the  cylinder,  and  that  if  this  could  be  the  case  the 
boiler  would  still  have  equal  safety  in  both  directions. 

In  whatever  direction,  therefore,  the  rolling  of  metal  gives  the 
greatest  tenacity,  in  the  same  direction  must  the  sheet  always  be 
bent  in  forming  the  convexity  of  the  cylinder.  It  follows  that  if 
we  suppose  the  tenacity  precisely  equal  in  both  directions,  the  lia- 
bility to  rupture  by  a  mere  internal  pressure  ought  to  be  twice  as 
great  along  the  longitudinal  direction  as  at  the  juncture  of  the  head.  - 
This  supposes  the  strain  regular,  and  the  riveting  not  to  weaken 
the  sheet. 

9.  To  know  how  large  we  may  safely  make  a  cylindrical  boiler, 
having  the  absolute  tenacity  of  the  metal,  in  the  strongest  direction, 
and  with  a  known  thickness,  we  have  only  to  revert  to  the  formula 

x  =  —.  •     That  is,  the  diameter  will  be  found  by  dividing  twice  the 
VOL.  II.— 9 


98  STRENGTH  OF  STEAM  BOILERS,  TUBES,  ETC. 

tenacity  by  the  greatest  force  per  unit  of  surface,  which  the  boiler  is 
ever  to  sustain. 

10.  When  knowing  the  absolute  tenacity  of  a  metal,  or  other 
material  reckoned  in  weight,  to  the  bar  of  a  given  area  in  its  cross 
section,  we  would  determine  the  thickness  of  that  metal  which  ought 
to  be  employed  in  a  boiler  of  given  diameter,  and  to  sustain  a  certain 

force,  we  may  use  the  formula  t  =  — ,  and  dividing  the  latter  num- 
ber of  this  equation  by  the  strength  of  the  square  bar,  which  we  may 
call  s,  we  obtain  the  thickness  demanded  in  the  direction  of  the  curve, 

which  we  may  denominate  p ;  so  that  p  =  ?L  ;  this  will  give  the 

thickness  of  the  boiler  plate  either  in  whole  numbers  or  decimals. 
Thus,  suppose  the  diameter  of  a  cylindrical  boiler  is  to  be  thirty-six 
inches — that  it  is  to  be  formed  of  iron  which  will  bear  55,000  Ibs.  to 
the  square  inch,  and  is  to  sustain  750  Ibs.  to  the  square  inch — what 
ought  to  be  the  thickness  of  the  metal  ? 

Here  x  =  36 
/=T50 

2*  =  110000,  consequently, 

p  =  86  X  T5°  =  .2454,  or  a  little  less  than  one-quarter  of  an 

inch. 

It  must,  however,  be  evident  that  the  minimum  tenacity  of  any 
particular  description  of  metal,  is  that  on  which  all  the  calculations 
ought  to  be  made  when  there  is  any  probability  that  the  actual  pres- 
sure will,  in  practice,  ever  reach  the  limit  assigned  as  the  value  of  f 
in  the  calculation. 

If  we  had  plates  of  diiferent  metals,  or  of  different  known  degrees 
of  tenacity  in  the  same  kind  of  metal,  and  were  desirous  of  ascer- 
taining how  strong  a  kind  we  must  employ  under  a  limited  thickness, 
diameter,  and.  pressure,  we  should  decide  the  point  by  transforming 

the  formula  p  =  J-,  into  ps  =  -^-,  and  then  into  s  =  — .    In  other 

terms,  in  order  to  know  the  strength  of  the  metal  required,  or  the 
direct  strain  which  an  inch  square  bar  of  the  same  ought  to  be  capable 
of  sustaining,  we  must  multiply  the  diameter  of  the  boiler  in  inches 
by  the  pressure  per  square  inch  in  pounds,  and  divide  the  product  by 
tivice  the  intended  thickness  in  parts  of  an  inch. 

Thus,  how  strong  a  metal  ought  to   be  employed  to  sustain  a 
'pressure  of  1000  Ibs.  to  the  square  inch,  in  a  boiler  thirty  inches  in 
diameter,  and  one-fourth  of  an  inch  thick  ? 

Here  s  =  80  X  10QO  =  60.000.     Hence  we  see  that  the  metal 
2  x  .25 

must  be  capable  of  sustaining  sixty  thousand  pounds  to  the  inch  bar, 
or  in  that  proportion  for  any  other  size.  This  formula  enables  us  to 
determine  whether  among  the  metals  of  known  tenacity,  any  one  can 
be  found  to  fulfil  the  conditions  under  the  thickness  assigned.] 


MACHINES.  99 


DIVISION     II. 


APPLICATION  OF  MECHANICS  TO  MACHINERY. 


INTRODUCTION. 

§  38.  Machines. — Machines  are  artificial  arrangements,  by  which 
forces  are  applied  to  produce  mechanical  effect.  Tools  or  instru- 
ments differ  from  machines  chiefly  in  their  being  applied  immediately 
to  the  work  to  be  done,  whilst  machines  are  intermediate. 

In  every  machine  we  have  to  distinguish  between  the  power  and 
the  resistance.  Power  is  the  cause  of  the  motion  of  the  machine, 
and  resistance  is  that  which  opposes  the  motion,  and  which  it  is  the 
object  of  the  machine  to  overcome.  The  powers  applied  to  machines 
are  modifications  of  those  supplied  by  nature  in  the  expansive  force 
of  heat,  the  action  of  gravity,  the  physical  force  of  men  and  animals, 
&c.  (Vol.  I.  §  60).  The  resistances  to  be  overcome  are  the  transport, 
and  the  change  of  form  and  texture  of  materials. 

There  are  in  every  machine  three  principal  parts.  One  which 
receives  the  power,  a  second  transmitting,  communicating,  or  modify- 
ing the  power,  and  a  third  applying  it.  In  the  common  flour  mill, 
considered  as  a  machine,  a  water  wheel  receives  the  power  of  a  water 
fall;  the  spur  wheel  and  pinion,  or  a  train  of  gear,  communicates 
the  motion  of  the  water  wheel  to  a  pair  of  stones  revolving  in  a  dif- 
ferent plane,  and  at  quite  different  speed  it  may  be,  from  that  of 
the  water  wheel,  and  these  stones  grind  the  corn,  or  do  the  work 
desired. 

Remark.  This  sub-division  is  not  always  manifest ;  for  there  are  machines,  in  which 
the  power  is  transmitted  so  directly  to  the  work  to  be  done,  that  the  communicators 
above  mentioned  are  not  apparent.  The  sub-division  is,  however,  convenient,  though  it 
would,  perhaps,  be  equally  so  to  apply  to  recipients  of  power,  the  generic  term  engine  or 
machine;  to  the  communicators  of  the  motion,  the  general  term  mechanism;  and  to  the 
parts  doing  the  work,  the  general  term  of  operators,  and  in  this  manner  to  consider  each 
separately,  as  they  are.  in  fact,  perfectly  distinct.  On  this  subject,  there  are  excellent 
observations  in  Willis'  "Principles  of  Mechanism,  1840,"  and  in  Ampere's  " Philotophie 
des  Sciences:'— TK. 

§  39.  Mechanical  Effect.— The  mechanical  effect  produced  by  a 
machine,  is  measured  by  the  work  done  in  a  given  time,  or  by  the 


100  USEFUL  AND  PREJUDICIAL  RESISTANCE. 

product  of  the  force  exerted,  and  the  distance  gone  through  in  a  unit 
of  time  in  the  direction  of  that  force.  If  P  be  the  force  exerted,  and 
*  the  distance  passed  through  in  a  second,  then  is  Ps  a  true  measure 
of  the  effect  of  the  machine  L  =  Ps  ft.  Ibs. 

It  is  very  usual  to  assume  a  somewhat  arbitrarily  chosen,  but  now 
pretty  generally  adopted  measure,  termed  horse  power,  as  the  unit 
of  mechanical  effect  of  engines  or  machines.  The  horse  power  is  in 
England  33,000  Ibs.  avoird.  raised  1  foot  high  in  a  minute.  This  is 
the  cheval  vapeur  of  the  French,  and  which  in  French  measures 
is  75  kilogrammes  raised  1  metre  high  in  a  second.  It  is  the 
PferdeTcraft  of  the  Germans,  or  510  Ibs.  Prussian,  raised  1  foot  high 
in  a  second. 

We  have  to  distinguish  the  useful  effect,  the  lost  effect,  and  the 
total  effect  of  machines.  The  useful  effect  is  the  work  done,  the  lost 
effect  is  that  consumed  in  overcoming  the  friction  of  the  parts  of  the 
machine  lost  in  shocks,  &c.,  and  the  total  effect  is  the  sum  of  these — 
the  effect  inherent  in  the  power,  or  the  effect  taken  out  of  it.  An 
engine  or  machine  is  so  much  the  more  perfect,  the  smaller  the  lost 
effect  compared  with  the  total  effect,  or  the  less  loss  there  arises  in 
adapting  and  transmitting  the  power.  The  ratio  of  the  useful  effect, 
produced  to  the  total  effect,  has  been  termed  the  efficiency  of  the 
machine.  If  L  =  the  total  effect  L^  —  the  useful  effect,  and  L2  =  the 

lost  effect,  the  efficiency  «?  =  — i  =      ~~    2 .     Thus,  the  more  per- 

A/  -L/ 

feet  the  machine,  the  more  nearly  its  efficiency  approaches  to  unity ; 
but  as  there  is  always  friction,  and  other  resistances  and  losses,  that 
degree  of  perfection  cannot  be  attained. 

Example.  An  ore  stamping  mill  consists  of  20  stampers,  each  of  which  weighs  250  Ibs., 
and  each  is  raised  40  times  per  minute,  1  foot  high.  The  machine  driving  these  is  a 
water  wheel,  taking  on  260  cubic  feet  per  minute,  and  the  fall  is  20  feet  high — required 
the  efficiency  of  this  machine.  The  useful  effect  is : 

20  .  _ '  4°  '  1  =  3333£  ft.  pounds  per  second  =  6  horse  power ;  the  total  effect, 

60 

however,  is :  26°  X  62.25  270  pounds  water  through  20  feet  per  second  =  5400  feet 
Ibs.  per  second  =  9,8  horse  power;  the  lost  effect  =  5400  —  3333£  =  2066$  feet 

Ibs.  =  3,75  horse  power;   and  the  efficiency  of  the  whole   arrangement  =  I 

5400 
=  0,62. 

§  40.  Useful  and  prejudicial  Resistance. — The  resistance  to  be 
overcome  by  machines  may  be  subdivided,  in  like  manner,  into 
useful  and  prejudicial  resistance,  but  as  the  power  is  applied  to 
the  useful  and  prejudicial  resistances  at  different  points,  we  cannot 
directly  set  the  power  equal  to  the  sum  of  the  useful  and  prejudi- 
cial resistance,  but  there  must  be  a  preliminary  reduction.  This 
reduction  is  made  by  means  of  the  spaces  simultaneously  passed 
through  by  the  different  points  of  resistance  of  the  machine.  If  the 
power  P  be  exerted  for  a  space  8,  and  the  useful  resistance  P1  for 
a  space  *„  and  the  prejudicial  resistance  P2  for  a  space  «2,  we  have 

Ps  =  PA  +  PA,  hence  P  =       P,  +       P2. 


USEFUL  AND  PREJUDICIAL  RESISTANCE. 


101 


The  point  in  the  machine  or  system  at  which  P  is  applied,  is 
termed  the  point  of  application  of  the  power,  and  the  points  at  which 
Pl  and  P2  act,  are  the  points  of  application  of  the  resistances;  we  have 

in  -l  Pj  the  useful  resistance  reduced  to  the  point  of  application  of 


power,  and  in  _*  P2,  the  prejudicial  resistance  reduced  to  the  same 

point.     The  power  is,  therefore,  equal  to  the  sum  of  the  useful  and 
prejudicial  resistances,  reduced  to  the  point  of  application  of  the 

power.     Again  Pl  =  —  P  --  ?  Pv  or  the  useful  resistance  is  equal 

si  8i 

to  the  difference  of  the  power  reduced  to  the  point  of  application  of 
that  resistance,  and  the  prejudicial  resistance  reduced  to  the  same 

point.    Hence  the  efficiency  of  a  machine:  /*  =     *Sl  =  -1  Px:  P=P1: 

—  P.  that  is,  the  quotient  of  the  useful  resistance  reduced  to  the 

«i 

power-point  and  the  power,  or  the  quotient  of  the  useful  resistance, 

and  the  power  reduced  to  the  point  of  application  of  the  useful 

resistance. 

Very  many  machines  are  adaptations  of  the  wheel  and  axle  (Vol.  I. 
§  152),  and  hence  the  reductions  may  often  be  accomplished  as  for 
a  lever.    If  in  the  wheel  and  axle  ABC,  Fig.  90, 
the  radius  of  the  wheel  CJ1  =  a,  the  drum's  ra-  Fis-  90- 

dius  CB  =  b,  then  the  statical  moment  of  the 
power  P,  =  Pa,  and  that  of  the  useful  resist- 
ance P1  =  Pjb,  and  therefore  the  useful  resist- 

ance reduced  to  the  power-point  A  =  -  Pv  and 

a 

the  power  reduced  to  the  point  of  application  b 

of  the  resistance  =  -  P.     If  the  prejudicial  re- 
o 

sistance  P2,  consist  in  the  axle  frictiony  (P+Pj 
+  G),  and  if  r  =  the  radius  DC  of  the  axle,  the 
moment  of  it  is  =  P2r,  and  therefore  the  preju- 
dicial resistance  reduced  to.  the  application  of 

power  =  ^L  =  fL  (P  +  P  +  G),  the  prejudi- 
ce a 

cial  resistance  reduced  to  the  point  of  applica- 
tion of  the  resistance 


Hence  P=  -P1=T 
a  a 

lastly,  ,  =  ^P1:P 

a 


£.  =  —  (P  +  Pl  +  G). 
6  o 

1+  G),alsoPt=  a-  P_^T 
bo 


Pa 


Example.  For  a  wheel  and  drum  weighing  250  Ibs  ,  the  wheel  being  30  inches  radius, 
Q* 


102  OF  THE  RIGIDITY  OF  CORDAGE. 

and  the  drum  6  inches  radius,  the  axle  $  inch  radius — the  useful  resistance  being  500 
Ibs.,  the  co-efficient  of  axle  friction  -j-'j,  then  the  useful  resistance  reduced  to  the  point  of 

application  of  the  power  =  _  P,  =  —  500  =  100  Ibs.,  and  the  prejudicial  resist- 
ance reduced  to  the  same  point : 

=  tL  (P  +  Pt  +  G)  =  J^  .       1       (750  +  P)  =  |  +  _?_, 

a  ff     2  .  30  T  600 

and  hence  we  have  to  put  the  power: 

P=100+*4--JL,  i.  e.  P  =  101,25  .  ^.=  101,42  Ibs., 

~  600  599 

and  the  efficiency  of  the  machine:  » 


101,42 
ON  THE  RIGIDITY  OF  CORDAGE. 

Amontons,  and,  after  him,  Coulomb,  experimented  on  the  rigidity 
of  hemp  ropes  and  cords:  and  Weisbach,  adopting  Coulomb's  method, 
has  recently  experimented  on  the  rigidity  of  hemp  and  wire  rope, 
such  as  are  used  in  the  drawing-shafts  of  mines. 

Coulomb  deduced  from  his  experiments,  that  the  law  of  this  resist- 
ance to  winding  may  be  represented  by  a  formula  composed  of  two 
terms ;  the  one,  a  constant  for  each  drum  or  pulley,  which  we  may 
designate  by  a,  and  which  the  distinguished  experimenter  termed 
"  natural  rigidity,"  because  it  depends  on  the  mode  of  manufacture 
of  the  rope,  and  on  the  degree  of  twist  given  to  the  threads  and 
strands;  the  other,  proportional  to  the  tension  Ton  the  rope,  and 
expressed  by  the  product  pT,  in  which  )3  is  a  constant  for  any  rope 
or  drum.  Thus  the  resistance  to  winding,  R  =  a  +  £  T. 

Coulomb  also  deduced  from  his  experiments  that  the  resistance  to 
winding  varies  inversely  as  the  diameter  d  of  the  drum  or  pulley ;  so 

that  -R  =  a  +  *—. 
d 

Naviet,  in  using  Coulomb's  experiments  to  construct  a  formula, 
assumed  that  the  co-efficients,  a  and  j3,  are  proportional  to  a  certain 
power  of  the  diameter,  depending  on  the  state  of  wear  of  the  rope; 
but  this  assumption  is  not  true.  For  it  would  lead  to  this,  that  a 
worn  rope  of  1  foot  diameter  has  the  same  rigidity  as  a  new  one, 
which  is  evidently  not  true  :  and  besides,  the  comparison  of  the  values 
of  a  and  j3  prove  that  the  power  to  which  the  diameter  has  to  be 
raised  cannot  be  the  same  for  the  two  terms  of  the  resistance. 

Coulomb's  experiments,  however,  show  that  the  rigidity  is  propor- 
tional to  the  number  n  of  threads  in  the  rope,  for  ropes  of  a  given 
manufacture. 

For   new  white  ropes,  the  formula : 

R  =  n[.0002  +  .000171  n  +  .000243  Q]  Ibs. 
for  drums  or  pulleys  of  1  foot  in  diameter,  and 

R  =  -  [.0002  +  .000171  n  +  .000243  Q]  Ibs. 
for  a  drum  of  diameter  d  in  feet,  accords  well  with  experiments.* 

*  For  the  complete  discussion  of  this  subject,  see  Morin,  "  Lecons  de  Mecanique 
pratique,"  lere  partie. 


WORKING  CONDITION.  103 

For  tarred  ropes:  R  =  ^[.001  +  .000232  n  +  .00028  Q]  Ibs. 

Whence  it  appears  that  tarred  ropes  are  rather  more  rigid  than  white 
ropes. 

Weisbach  has  deduced  from  his  experiments  on  wire  rope,  (4  wires 
round  a  core  in  each  strand,  and  4  strands  round  a  core  in  the  rope,) 
weighing  3  Ibs.  to  the  fathom,  the  formula : 

R  =  0,72  +  0,0262  Q  Ibs., 
d 

in  which  Q  is  the  strain  on  the  rope  in  cwts.,  and  d  the  diameter  of 
the  pulley.     Whereas,  for  the  hemp  ropes,  jit  for  the  same  uses,  or 

of  the  same  strength,  jR  =  3,02  +  0,086  — :  or  the  rigidity  is  con- 

d 
siderably  greater. 

Wire  ropes,  newly  tarred  or  greased,  have  about  40  per  cent,  less 
rigidity  than  untarred  ropes.* 

§  41.  Working  Condition. — When  a  machine  is  set  in  motion,  it  soon 
comes  to  its  working  condition,  that  is,  there  recur  at  regular  periods 
the  same  relative  position  of  the  parts,  the  periodic  motion  becomes 
uniformly  so.  In  this  condition  we  assume  machines  to  be  in  apply- 
ing our  principles,  but  their  working  condition  may,  according  to 
circumstances,  be  either  uniform  or  variable.  The  causes  inducing 
irregularity  are  variations  in  the  power  or  in  the  resistance,  as  also 
the  proportions  of,  or  construction  of  the  machine,  in  reference  to 
variations  in  the  spaces  described  in  a  given  time  by  the  power  and 
resistance,  and  the  state  of  motion  of  inert  masses. 

In  a  steam-engine,  the  power  is  variable  when  the  engine  "  works 
expansively,"  that  is,  when  the  steam  is  cut  off 
during  the  progressive  motion  of  the  piston.  Fig-  91- 

In  a  mill  for  rolling  iron,  the  power  and  resist- 
ance are  continually  varying,  because  the  forge 
hammer  is  out  of  gear  when  falling  on  the 
blooms,  and,  therefore,  the  working  condition 
of  the  machines  is  irregular.  If  the  engine 
work  expansively,  then  there  would  arise  from 
the  combination  of  the  engine,  and  hammer, 
and  rollers,  three  causes  of  irregularity.  When 
a  weight  G,  Fig.  91,  is  raised  by  a  steam- 
engine  with  uniform  pressure  by  means  of  a 
wheel  CJJ0,  and  crank  CBQ,  the  machine  has 
a  variable  working  condition,  because  equal 
spaces  JfAv  Jl^,  A^,  J13J14,  of  the  resist- 
ance correspond  to  very  unequal  distances 
described  by  the  power,  and,  therefore,  the 
ratio  during  a  half  revolution  is  variable,  but 
for  periods  of  a  half  revolution  it  is  uniform. 

*  Weisbach's  paper  on  this  subject  is  contained  in  the  first  number  of  a  journal  pub- 
lished at  Freyberg,  under  the  title  "  Der  Ingenieur  Zeitschrifl  fur  das  gesammte  Inge- 
nieurwesen,"  1846. 


104  WORKING  CONDITION. 

In  the  case  of  uniform  working  condition,  the  inert  masses  on  a 
machine  are  without  influence,  because  it  is  only  at  first,  when  the 
machine  is  still  accelerating  in  motion,  that  they  absorb  mechanical 
effect,  but  later,  when  uniform  motion  has  established  itself,  there  is 
neither  loss  nor  gain  of  mechanical  effect  (Vol.  I.  §  52).  But  if,  on 
the  other  hand,  a  machine  be  subject  to  irregular  working  conditions, 
the  inert  masses  of  the  parts  have  an  essential  influence  on  the  motion 
of  the  machine,  because  they  absorb  mechanical  effect  at  every  acce- 
leration of  speed,  and  this  they  again  give  off  at  each  retardation. 
If  M  be  the  sum  of  all  the  masses  reduced  to  the  power  or  resistance- 
point  of  a  machine,  vl  and  v2,  the  minimum  and  maximum  velocities 
of  the  power,  or  resistance-points,  we  have  the  mechanical  effect 
which  the  inert  masses  absorb  during  their  transition  from  the 
velocity  vl  to  v2,  and  which  they  again  give  out  in  passing  from  v2 

to  Vl  =  (v*  ~tl  \M.     Thus,  in  each   period,  the  inertia  of  the 

masses  increases  and  diminishes  the  lost  effect  by  the  above  amount, 
and,  therefore,  the  total  effect  for  the  whole  period,  or  the  mean 
effect  is  the  same  as  if  these  inert  masses  were  not  there.  Hence, 
as  a  general  formula,  Ps  =  P^  +  P2*2  holds  good  for  a  variable 
working  condition,  if  by  «,  *„  «2,  we  understand  the  spaces  described 
in  a  complete  period,  or  if  for  P,  Pv  Pv  we  substitute  the  mean 
values  of  the  power,  and  useful  and  prejudicial  resistance,  for  a 
given  period.  For  the  case  of  accelerating  motion :  Ps  =  P  s  + 

PA  +  (l\  M,  hence  „,-*,  *~(*A±jW      This  for- 


mula  shows  that  the  variations  of  velocity  of  a  machine  are  not  only 
less,  the  less  the  difference  between  the  effects  of  the  power  and  the 
sum  of  the  effects  of  the  resistances,  but  also  the  greater  the  masses 
:  the  parts  of  the  machine,  and  the  greater  their  velocity. 

tfmar*.  It  does  not  follow  that  because  the  mass  of  the  parts  do  not  affect  the  efficiency 
•achine,  but  only  its  working  condition,  that  it  is  a  matter  of  indifference,  whether 
.arts  of  a  machine  have  more  or  less  mass.     Weight  increases  friction,  gives  rise  to 
shocks,  &c.,  which  are  prejud.cial.     But  of  this  in  the  sequel 


COMMON  BALANCE.  105 


SECTION  II. 


OF  MOVING  POWERS  AND  THEIR  RECIPIENT  MACHINES. 


CHAPTER  I. 

OF  THE  MEASURE  OF  POWERS  AND  THEIR  EFFECTS. 

§  42.  Dynamometer. — In  order  to  determine  the  mechanical  effect 
produced  by  powers  and  machines,  in  terms  of  the  horse  power  unit, 
three  elements  are  necessary,  viz :  The  magnitude  of  the  power  or 
effort,  the  distance  passed  through  by  it,  and  the  time  during  which 
the  power  has  acted. 

To  enable  us  to  represent  the  effect  of  forces,  we  must,  therefore, 
have  measures  of  the  force  applied,  the  distance,  and  the  time.  Dy- 
namometers serve  to  measure  the  force  applied,  the  chain  is  generally 
used  for  measuring  space,  and  clocks  or  watches  measure  time.  If  P 
be  the  magnitude  of  the  force  indicated  by  the  dynamometer,  and  s 
the  distance  throughout  which  it  has  acted  during  the  time  t,  then, 

the  work  or  mechanical  effect  produced  in  this  time  is  =  Ps,  and  the 

D- 
work  per  second  L  =  — 

Of  dynamometers,  there  are  various  forms.  The  common  balance 
is  a  dynamometer,  and  is  used  to  measure  the  force  of  gravity  or 
weight.  Modifications  of  spring  balances,  and  the  friction  brake  are 
the  dynamometers  applied  to  measure  forces  producing  mechanical 
effect.  The  friction  brake  is  applied  to  measuring  the  mechanical 
effect  given  off  by  revolving  axles. 

Balances  are  simple  or  compound  levers,  on  which  the  force  or 
weight  to  be  measured  is  set  in  equilibrium  with  standard  weights. 
Balances  are  either  equal  or  unequal-armed  levers,  and  the  latter 
are  variously  combined,  according  to  the  purposes  to  which  they  are 
applied. 

§  43.  Common  Balance. — The  common  balance  is  a  lever  with 
equal  arms,  Fig.  92,  on  which  the  weight  Q  to  be  measured  is 
equilibriated  by  an  equal  weight  P.  JIB  is  the  beam  with  its  points 
of  suspension,  (Fr.  fleau,  Ger.  Waagebalken,}  CD  the  index  or 


106 


COMMON  BALANCE. 


Fig.  92.  point,  (Fr.  aiguille,  Ger.  Zunge,} 

CE  the  support  or  fork,  (Fr.  sup- 
port, Ger.  Scheere,)  C  is  the  knife- 
edge  or  fulcrum,  a  three -sided 
prism  of  hard  steel. 

The  requisites  of  a  balance  are : 

1.  That  it  shall  take  a  horizontal 
position  when  the  weights  in  the 
two  scales  are  equal,  and  only  then. 

2.  The  balance  must  have  sensi- 
bility and  stability,  that  is,  it  must 
play  with  a  very  slight  difference 
of  weight  in  either  of  the  scales, 
and  must  readily  recover  its  hori- 
zontal position,  when  the  weights 
are  again  made  equal. 

That  a  balance  with  equal  weights  in  the  two  scales  may  be  in  ad- 
justment, the  arms  must  be  perfectly  equal.  If  a  be  the  length  of 
the  one,  and  b  be  that  of  the  other  arm ;  P  the  weight  in  the  one 
scale,  and  Q  that  in  the  other.  Then  when  the  beam  is  horizontal 
Pa  =  Qb.  If,  however,  we  transpose  the  weights  P  and  Q,  we  have 
again  Pb  =  Qa,  if  the  beam  retain  its  horizontal  position.  From 
the  two  equations  we  have  P2  ab  =  Q2  ab,  therefore,  P  =  Q,  and 
likewise  a  =  b.  "When,  therefore,  on  transposing  the  weights,  the 
equilibrium  is  not  disturbed,  it  is  a  test  of  the  truth  of  the  balance. 
A  balance  may  also  be  tested  in  the  following  manner.  If  we  put 
one  after  the  other  two  weights  P  and  P  into  equilibrium  with  a  third 
Q  in  the  opposite  scale,  the  two  weights  P  and  P  are  equal  to  each 
other  though  not  necessarily  equal  to  Q.  If,  then,  we  lay  the  two 
equal  weights  in  the  opposite  scales,  removing  Q,  we  should  have  in 
case  of  equilibrium  Pa  =  Pb,  and  hence  a  =  b.  Thus  the  hori- 


SENSIBILITY  OF  BALANCES. 


107 


zontality  of  the  balance  when  two  equal  weights  are  laid  upon  it,  is 
a  direct  proof  of  its  truth  or  justness.  Small  inaccuracies  may 
be  adjusted  by  means  of  the  screws  K,  L,  as  shown  on  the  balance 
(Fig.  93),  which  serves  to  press  out  or  putt  in  the  points  of  sus- 
pension. 

If  a  balance  indicates  weights  P  and  Q  for  the  same  body,  accord- 
ing as  it  has.  been  weighed  in  the  one  scale  or  the  other,  we  have 
for  the  true  weight  X  of  that  body  :  Xa  =  Pb  and  Xb  =  Qa,  hence 
X2  .  ab  =  PQ  .  ab,  or,  X3  =  PQ,  and  X  =  >/PQ,  or  the  geometric 
mean  between  the  two  values  is  the  true  weight  of  the  body. 


We  may  also  put  X  =  V  P  (P  +  Q  —  P) 
~ 


P      l  + 


and 


approximately  =  P  M  +       ~     )  = 


J 
j  when,  as  is  usual,  the 


difference  Q  —  P  is  small  ;  we  may,  therefore,  take  the  arithmetical 
mean  of  the  two  weighings  as  the  true  weight. 

§  44.  Sensibility  of  Balances.  —  That  the  balance  may  move  as 
freely  as  possible,  and  particularly  that  it  may  not  be  retarded  by 
friction  at  the  fulcrum,  this  is  formed  into  a  three-sided  prism  or 
knife-edge  of  steel,  and  it  rests  on  hardened  steel  plates,  or  on  agate, 
or  other  stone.  In  order,  further,  that  the  direction  of  the  resultant 
of  the  loaded  or  empty  scale  may  pass  through  the  point  of  sus- 
pension uninfluenced  by  friction,  in  order,  in  short,  that  the  leverage 
of  the  scale  may  remain  constant,  it  is  necessary  to  hang  the  scales 
by  knife-edges.  In  whatever  manner  such  a  balance  is  loaded,  we  may 
always  assume  that  the  weights  act  at  the  points  of  suspension,  and 
that  the  points  of  application  of  the  resultant  of  these  two  forces  is  in 
the  line  joining  the  points 

of  suspension.    As,  accord-  Fig-  94- 

ing  to  Vol.  I.  §  122,  a  sus- 
pended body  is  only  in 
equilibrium  when  its  centre 


of  suspension,  it  is  evident 

that  the  fulcrum  D  of  the 

balance,  Fig.  94,  should  be 

above  the  centre  of  gravity 

S  of  the  empty  beam,  and 

also  not  below  the  line  JIB 

drawn  through  the    points 

of  suspension.     In  what  follows,  we  shall  assume  that  the  fulcrum 

D  is  above  AB  and  above  S. 

The  deviation  of  a  balance  from  horizontality  is  the  measure  of  its 
sensibility,  and  we  have  to  investigate  the  dependence  of  this  on  the 
difference  of  weight  in  the  scales.  If,  for  this,  we  put  the  length  of 
the  arms  CA  and  CB  =  Z,  the  distance  CD  of  the  fulcrum  from  the 
line  passing  through  the  points  of  suspension  =  a,  the  distance  SD 
of  the  centre  of  gravity  from  the  fulcrum  =  s,  if  further  we  put  the 


108  SENSIBILITY  OF  BALANCES. 

angle  of  deviation  from  the  horizontal  =  <?>,  the  weight  of  the  empty 
beam  =  G,  the  weight  on  the  one  side  =  P,  and  that  on  the  other 
=  P  +  Z,  or  the  difference  =  Z,  and,  lastly,  the  weight  of  each 
scale,  and  its  apurtenances  =  Q,  we  have  the  statical  moment  on 
the  one  side  of  the  balance  :  (P  -f  Q  +  Z)  .  DH=  (P  +  Q  +  Z) 
(CK—DE)  =  (P  +  Q  4-  Z)  (I  cos.  f  —  asin.  1>)  and  on  the  other 
side  :  (P  +  Q)  .  DL  +  G  .  DF=  (P  +  Q)  (CM  +  DE)  +  G  .  DF 
=  (p  _}.  Q)  (I  cos.  $+a  sin.  <j>)  +  G  s  sin.  <|>;  therefore,  equilibrium  : 
(P  +  Q  -4-  Z)  I  cos.  t  —  a  sin.  *)  =  (P  +  Q)  (I  cos.  ?  +  «  *m.  <J>) 
+  G  8  sin.  t,  or,  if  we  introduce  tang.  •}>,  and  transform  : 
[[2  (P  +  Q)  -f  Z]  a  +  G  *]  taw#.  $>  =  Z  I,  therefore, 


This  expression  informs  us  that  the  deviation,  and,  therefore,  the 
sensibility  of  the  balance,  increases  with  the  length  of  the  beam,  and 
decreases  as  the  distances  a  and  s  increase.  Again,  a  heavy  balance 
is,  cceteris  paribus,  less  sensible  than  a  light  one,  and  the  sensibility 
decreases  continually,  the  greater  the  weights  put  upon  the  scales. 
In  order  to  increase  the  sensibility  of  a  balance,  the  line  J1R  joining 
the  points  of  suspension  and  the  centre  of  gravity  of  the  balance, 
must  be  brought  nearer  to  each  other. 

If  a  and  s  were  equal  to  0,  or  if  the  points  D  and  S  were  in  the 

Zl 
line  JIB,  we  should  have  tang.  <j>  =  —  =00,  therefore  <j>  =  90°  ;  and 

o 

therefore  the  slightest  difference  of  weights  would  make  the  beam 
kick  or  deflect  90°.     In  this  case  for  Z  =  0,  we  should  have  : 

tang.  $  =  -,  i.  e.  the  beam  would  be  at  rest  in  any  position,  if  the 

weights  were  equal  in  each  scale,  and  the  balance  would  therefore 
be  useless.     If  we  make  only  a  =  0,  or  put  the  fulcrum  in  the  line 

71 
.iB,  then  tang.  <j>  =  _  ,  or  the  sensibility  is  independent  of  the 

amount  weighed  by  the  balance.     By  means  of  a  counterweight  N 
with  a  screw  adjustment,  Fig.  93,  the  sensibility  may  be  regulated. 

§  45.  Stability  and  Motion  of  Balances.  —  The  stability,  or  statical 
moment,  with  which  a  balance  with  equal  weights  returns  to  the  po- 
sition of  equilibrium,  when  it  has  inclined  by  an  angle  <j>,  is  deter- 
mined by  the  formula: 

S=  2  (P  +  Q)  .  DE  +  G  .  DF=  [2  (P  +  Q)  a  +  Gs]  sin.  ?. 
Hence,  the  measure  of  stability  increases  with  the  weights  P,  Q,  and 
G,  and  with  the  distances  a  and  *,  but  is  independent  of  the  length 
of  the  beam. 

A  balance  vibrating  may  be  compared  with  a  pendulum,  and  the 
time  of  its  vibrations  may  be  determined  by  the  theory  of  the  pen- 
dulum. 2  (P  +  Q)  a  is  the  statical  moment,  and  2  (P  +  Q)  .  rfD2 
=  2  (P  +  Q)  (I2  +  a2}  is  the  moment  of  inertia  of  the  loaded  scales, 
and  Gs  is  the  statical  moment  of  the  empty  beam.  If  we  put  the 
moment  of  inertia  'of  the  beam  =  G#2,  we  have  for  the  length  of 


UNEQUAL-ARMED  BALANCES.  109 

the  pendulum  which  would  be  isochronous  with  the  balance  (Vol   I 

§  250): 


2(P+Q)a+Gs 
and  hence  the  time  of  vibration  of  the  balance : 


for  which,  when  a  is  very  small  or  0,  we  may  put: 

' 


f 
\ 


_ 

It  is  evident  from  this  that  the  time  of  a  vibration  increases  as 
P,  Q,  /  increase,  and  as  a  and  s  diminish.  Therefore,  with  equal 
weights,  a  balance  vibrates  the  more  slowly,  the  more  sensible  it  is, 
and  therefore  weighing  by  a  sensible  balance,  is  a  slower  process 
than  with  a  less  sensible  one.  On  this  account  it  is  useful  to  furnish 
sensible  balances  with  divided  scales  (as  Z,  Fig.  93).  In  order  to 
judge  of  the  indication  of  such  a  scale,  let  us  put  Z  the  additional 
weight  =  0  in  the  denominator  of  the  formula  : 

Zl 


=  [2(P+Q)+Z]a+Gs' 
and  write  <j>  instead  of  tang.  $,  we  then  get: 

,  =  Zl 

2  (P  +  Q)  a  +  G  s 

If  we  then  put  Z  for  Zv  and  <j>  for  ^  we  get: 


i  ~  Z  :  Zi;  or'  for  sma11 


ferences  of  weight,  the  angle  of  deviation  is  proportional  to  that 
difference.  Hence,  again  <|>:<J>1  —  $  =  Z  :  Zl  —  Z;  and  therefore 

Z  =  _  -  —  (Zj  —  Z).     We  can,  therefore,  find  the  difference  of 

$1  —  * 

weights  corresponding  to  a  deviation  $,  by  trying  by  hotf  much  the 
deviation  is  increased,  when  the  difference  of  weights  is  increased 
by  a  given  small  quantity,  and  then  multiplying  this  increase 
(Zx  —  Z)  by  the  ratio  of  the  first  deviation  to  the  greater  deviation 
obtained. 

Remark.  Balances  such  as  we  have  been  considering,  are  used  of  all  dimensions,  and 
of  all  degrees  of  delicacy  and  perfection.  Fig.  92  is  the  usual  form  of  this  balance  used 
in  trade,  and  Fig.  93  represents  the  balances  used  in  assaying,  analysis,  and  in  physical 
researches.  .  Such  balances  as  Fig.  93,  are  adapted  to  weigh  not  more  than  1  lb.;  but 

they  will  turn  with  -L  of  a  grain,  or  with  -  !  -  of  a  pound.     The  finest  balances  that 
50  350000 

have  been  made,  render  _  _  _  part  of  the  weight  appreciable,  but  such  balances  are 

1,000,000 

only  for  extremely  delicate  work.  Even  large  balances  may  be  constructed  with  a  very 
high  degree  of  sensibility.  For  minute  details  on  this  subject,  the  student  is  referred  to 
Lardner's  and  Kater's  Mechanics,  in  "  Lardner's  Cyclopaedia." 

§  46.  Unequal-armed  Balances.—  The  balance  with  unequal  arms, 
termed  statira,  or  Roman  balance  and  steelyard,  presents  itself  m 
VOL.  II.—  10 


110 


UNEQUAL-ARMED  BALANCES. 


three  different  forms,  viz:  steel- 
Fig.  95.  yard  with  movable  weight,  steel- 
yard with  proportional  weights, 
and  steelyard  with  fixed  weight. 
The  steelyard  with  running 
weight  is  a  lever  with  unequal 
arms  AB,  Fig.  95,  on  the  shorter 
arm  of  which  C*fl,  a  scale  is  sus- 
pended, and  on  the  longer  di- 
vided arm  of  which  there  is  a 
running  weight,  which  can  be 
brought  into  equilibrium  with 
the  body  to  be  weighed  Q.  If 
10  be  the  leverage  CO  of  the 

running  weight  G,  when  it  balances  the  empty  scale,  we  have  for 
the  statical  moment  with  which  the  empty  scale  acts  X0  =  G?0,  but 
if  ln  =  the  leverage  CG,  with  which  the  running  weight  balances  a 
certain  weight  Q,  we  have  for  its  statical  moment :  Xn  =  Gln ;  and 
hence,  by  subtraction,  the  moment  of  the  weight  Q,  =  Xn  —  XQ 
=  G  (ln—  10)  =  G  .  OG.  If  again  a  =  Otf,  the  leverage  of  the 
weight,  and  if  b  be  the  distance  OG  of  the  running  weight  from  the 
point  0,  at  which  it  balances  the  empty  scale,  we  have  Qa  =  G6, 

/•> 

and  therefore  the  weight  Q  =  —  b.    Hence  the  weight  Q  of  the  body 

to  be  weighed  is  proportional  to  b  the  distance  of  the  running  weight 
from  the  point  0.  26  corresponds  to  2  Q,  3  6  to  3  Q,  &c.  And, 
therefore,  the  scale  OB  is  to  be  divided  into  equal  parts,  starting 
from  0.  The  unit  of  division  is  obtained  by  trying  what  weight 
Qn  must  be  put  on  the  scale  to  balance  G,  placed  at  the  end  B. 

S\  T) 

Then  Qn  is  the  number  of  division,  and  therefore the  scale  or 

unit  of  division  of  OB.  If,  for  example,  the  running  weight  is  at  J5, 
when  Q  =  100  Ibs.,  then  OB  must  be  divided  into  100  equal  parts, 

and,  therefore,  the  unit  of  the 
Fig.  96.  OB 

scale=_ 

weight  Q,  the  weight  G  has 
to  be  placed  at  a  distance  6 
=  80,  to  adjust  the  balance, 
then  Q  =  80  Ibs.;  and  so 
on. 

In  the  steelyard,  Fig.  96, 
with  proportional  weights, 
the  body  to  be  weighed  hangs 
on  the  shorter,  and  the  stand- 
ard weights  are  put  on  the 


If  for  another 


longer  arm. 


C'K 

The  ratio  —  — 
CJi 


WEIGH-BRIDGES. 


Ill 


=-  of  the  arms  is  generally  simple,  as  \°,  jn  ^{^  case  the  balance 

becomes  a  decimal  balance.  If  the  balance  has  been  brought  to 
adjustment  or  horizontality  by  a  standard  weight,  then  for  the  weight 

Q  of  the  body  in  the  scale,  we  haveQa=  Gb;  hence  Q=-  G,  and 

a 

therefore  the  weight  of  the  merchandise  is  found  by  multiplying  the 
small  weight  G  by  a  constant  number,  for  instance,  10  in  the  deci- 
mal balance,  or  if  the  latter  be  assumed  -  times  as  heavy  as  it  really  is. 

The  steelyard  with  fixed  weight,  Fig.  97,  called  the  Danish  ba- 
lance, has  a  movable  fulcrum  C,  (or  it  is  movable  on  its  fulcrum,) 
which  can  be  placed  at  any  point  in  the  length  of  the  lever,  so  as 
to  balance  the  weights  Q  hung  on  one  end,  by  the  constant  weight 
G,  fixed  at  the  other. 

The  divisions  in  this  case  are  1?'e-  97> 

unequal,  as  will  appear  in  the 
following  remark: — 

Remark.  In  order  to  divide  the  Danish 
steelyard,  Fig.  98,  draw  through  its  centre 
of  gravity  S  and  its  point  for  suspension 
B  two  parallel  lines,  and  set  off  on  these 
from  S  arid  B  equal  divisions,  and  draw 
from  the  first  point  of  division  of  the  one, 
••lines  to  the  points  of  division  I,  II,  III, 
IV  of  the    other  parallel    straight   line. 
These  lines  cut  the  axis  BS  of  the  beam 
in  the  points  of  division  required.     The 
point  of  intersection  (1)  of  the  line  I,  I, 
bisects  SB,  and,  by  placing  the  fulcrum 
there,  the  weight  of  the  merchandise  is  equal  to  the  total  weight  of  the  steelyard,  if  it 
be  horizontal,  or  in  a  state  of  equilibrium.     The 
point  of  intersection  (2)  in  the  line  I,  II,  is  as  far 
again  from   S  as  from  B]  and,  therefore,  when 
this  point  is  supported  Q  =  2  G,  when  equili 
brium  is  established  similarly,  the  point  of  divi 
sion  (3)  in  the  line  I,  III,  is  3  times  as  far  from 
S  as  from  JB;  and  hence  for  Q  =  3  G,  the  ful 
crum  must  be  moved  to  this  point.    It  is  also  evi 
dent  that  by  supporting  the  points  of  division  £ 
5,  &c.,  the  weight  Q  =  £  G,  3  G,  and  so  on,  when 
the  beam  is  in  a  state  of  equilibrium.     We  see 
from  this  that  the  points  of  division  lie  nearer 
together  for  heavy  weights,  and  further  apart  for 
light  weights,  and  that,  therefore,  the  sensibility 
of  this  balance  is  very  variable. 

§  47.  Weigh-bridges. — Compound  balances  consist  of  two,  three, 
or  more  simple  levers,  and  are  chiefly  used  as  weighing-tables  or 
weigh-bridges  for  carts,  wagons,  animals,  &c.  Being  used  for  weigh- 
ing great  weights,  they  are  generally  proportional  balances.  The 
scale  of  the  ordinary  steelyard  is  replaced  here  by  a  floor,  which 
should  be  so  supported,  and  connected  with  the  levers,  that  the 
receiving  and  removing  of  the  body  to  be  weighed  may  be  as  conve- 
niently done  as  possible,  and  that  the  indication  of  the  balance  may 
be  independent  of  the  position  of  the  body  on  the  floor. 


112 


WEIGH-BRIDGES. 


Fig.  99  represents  a  very  good  kind  of  weigh-bridge  by  Schwilgue 
in  Strasbourg  (balance  a  bascule).     This  weigh-bridge  consists  of  a 

Fig.  99. 


two-armed  lever  ACB,  of  a  simple  single-armed  lever  A^C^^  and 
of  two  fork-like  single-armed  levers  BjSv  DS2,  &c.  The  fulcrum  of 
these  levers  are  C,  Cl  and  D^D^  The  bridge  or  floor  W  is  only  par- 
tially shown,  and  only  one  of  the  fork-formed  levers  is  visible.  The 
bridge  usually  rests  on  four  bolts  Kv  Kz,  &c.,  but  during  the  weigh- 
ing of  any  body,  it  is  supported  on  the  four  knife-edges  S^  S2,  &c., 
attached  to  the  fork-shaped  levers.  In  order  to  do  this,  the  support 
E  of  the  balance  AB  is  movable  up  and  down  by  means  of  a  pinion 
and  rack  (not  visible  in  the  drawing).  The  business  of  weighing  con- 
sists in  raising  the  support  EC,  when  the  wagon  has  been  brought  on 
to  the  floor,  in  adjusting  the  weight  G  in  the  scale,  and  finally  in 
lowering  the  bridge  "on  to  its  bearings  K^  K^  &c.  The  usual  propor- 
tions of  the  levers  are:  — —  =  2,  _ -1  =  5,  and  the  arms  — -1  =  10. 

If,  therefore,  the  empty  balance  has  been  adjusted,  the  force  at  -B 
or  Al  =  2  G,  the  force  at  Bl  =  5  times  the  force  in  A  =  10  G,  and 
lastly,  the  force  in  S  =  10  times  that  in  Bl  =  100  G.  And,  there- 
fore, when  equilibrium  is  established,  the  weight  on  the  floor  is  100 
times  that  laid  on  the  scale  at  G,  and  this  makes  a  centesimal  scale. 
Another  form  of  weigh-bridge  such  as  is  constructed  at  Angers,  is 
shown  in  Fig.  100.  The  bridge  Wof  this  balance,  rests  by  means  of 

Fig.  100. 


PORTABLE  WEIGH-BRIDGE. 


113 


four  supports  at  J515  B2,  &c.,  on  the  fork-shaped  single-armed  levers 
^1B1CV  J12B2C2,  of  which  the  latter  is  connected  with  the  prolonga- 
tion Cfl  of  the  former,  by  a  lever  DEF  with  equal  arms.  Until 
the  bridge  is  to  be  used,  it  rests  on  beams  S,  S,  but  when  the  load  is 
brought  on,  the  support  LL  of  the  balance  JIB  is  raised  (and  with 
it  the  whole  system  of  levers),  by  means  of  a  pinion  and  rack-work, 
and  then  so  much  weight  is  laid  in  the  scale  at  G,  as  is  necessary  to 
produce  equilibrium. 

In  whatever  manner  the  weight  Q  is  set  upon  the  bridge  W,  the 
sum  of  the  forces  at  Bv  -B2,  &c.,  is  always  equal  to  that  weight. 

But  the  ratio    2    2  is  equal  to  the  ratio     1    *  =  5i  of  the  length  of  the 
C2B2  C1B1       6j 

arms,  and  the  length  of  the  arm  DE  =  length  of  arm  DF,  as 
also  C^H  =  C^.  It  is,  therefore,  the  same  in  effect,  whether  a 
part  of  the  weight  Q  is  taken  up  on  J52,  or  directly  on  5X  ;  or  the 
conditions  of  equilibrium  of  the  lever  C1B1J11  are  the  same,  whether 
the  whole  weight  Q  act  directly  on  Bv  or  only  a  part  of  it  in  B^  and 
the  rest  in  B2,  and  only  transferred  by  the  levers  C2B2A2,  EDF  and 


CHto 


If,  further,  "  be  the  ratio  of  the  arms 

o  CB 


of   the 


upper  balance,  the  force  on  the  connecting  rod  BJil  =  _  .  G,  and 
'hence  the  weight  on  the  floor  supposed  previously  adjusted,  is : 

Q  =  £!i  .  a.  G.    Generally  -  =  £  =  Y>,  hence,  ^  =  '  £°,  and   the 

6j     o  o       6j  £ 

balance  is  centesimal. 

§  48.  Portable  Weigh-bridge.  —  In  factories  and  warehouses, 
various  forms  and  dimensions  of  weighing-tables,  after  the  design 
of  that  of  Quintenz,  are  used.  This  balance,  which  is  represented 
in  Fig.  101,  consists  of  three  levers  .^CD,  EF,  and  HK.  On  the  first 
lever  hangs  the  scale-pan  G,  for  the  weights,  and  two  rods  DE  and  BH. 

Fig.  101. 


114:  PORTABLE  WEIGH-BRIDGE. 

The  rod  DE  carries  the  lever  turning  on  the  fixed  point  F,  and  the 
second  rod  BH  carries  the  lever  HK,  the  fulcrum  of  which  rests  upon 
the  lever  EF.  In  order  to  provide  a  safe  position  for  the  two  latter 
levers,  they  are  made  fork-shaped,  and  the  axes  F  and  K  on  which 
they  turn,  are  formed  by  the  two  knife-edges.  On  the  lever  HK, 
the  trapezoidal  floor  ML  is  placed  to  receive  the  loads  to  be  weighed, 
and  it  is  provided  with  a  back-board  JWJV,  which  protects  the  more 
delicate  parts  of  the  balance  from  injury.  Before  and  after  the  act 
of  weighing,  the  lever  formed  by  the  border  of  the  floor  rests  on 
three  points,  of  which  only  one,  R,  is  visible  in  our  section  ;  and  the 
balance  beam  JID  is  supported  by  a  lever-formed  catch  S,  provided 
with  a  handle.  When  the  merchandise  is  laid  on  the  table,  the 
catch  is  put  down,  and  weights  are  laid  on  at  G,  till  the  balance  AD 
is  in  adjustment.  The  catch  is  again  raised,  so  that  HK  comes 
again  to  bear  on  the  three  points,  and  the  merchandise  can  be  re- 
moved without  injury  to  the  balance.  The  balance  JID  is  known  to 
be  horizontal  by  an  index  Z,  and  the  empty  balance  is  adjusted  by 
a  small  movable  weight  T,  or  by  a  special  adjusting  weight  laid  on 
the  scale  at  G. 

In  this,  as  in  other  weigh-bridges,  it  is  necessary  that  its  indica- 
tions be  independent  of  the  manner  in  which  the  goods  are  placed 
upon  the  table  or  floor.  That  this  condition  may  be  satisfied,  it  is 

EF 

necessary  tjiat  the  ratio  —  —  of  the  arms  of  the  lever  EKF,  be  equal 
KF 

CD 

to  the  ratio  —  —  of  the  arms  of  the  balance  beam  JID.     A  part  X  of 
CB 

the  weight  Q  on  the  floor  is  transferred  by  the  connecting  rod  BH 
to  the  balance  beam  AH,  and  acts  on  this  with  the  statical  moment, 

CB  .  X  ;  another  part  F,  goes  through  K  to  the  lever  EF,  and  acts 

KF 
at  E  with  the  force  -  .  F.     But  this"  force  goes  by  means  of  the 

rod  DE  to  D  to  act  on  the  balance  beam.     The  part  F,  therefore, 

_      jrp 

acts  with  the  statical  moment,  CD  .  —^  .  Y,  and  at  B  with  the  force 

Er 

C*T)       W 

-  .  —  —  .  F  on  the  balance  beam  AD.  That  the  equilibrium  of 
CB  Er 

the  balance  beam  may  not  depend  on  either  X  or  F  alone,  but  on 
the  sum  of  them  Q  =  X  +  F,  it  is  requisite  that  F  should  act  on  the 
point  B,  exactly  as  if  it  were  applied  there  directly,  or  that  : 
CD     KF     v      v  .        CD     KF       n     ,       ,        CD       EF      T, 

" 


CB  '  EF  '   •       CS  '  EF        '  '  CB       KF~ 

we  denote  the  arms  CA  and  CB  by  a  and  6,  we  have  here  as  in  the 
simple  balance   Ga  =  (X  +  F)  b  =  Q  6,  and,  therefore,  the  weight 

required  Q=  -  G,  for  example,  =  10  G,  if  the  length  CB  is  =  ^  of 
o 

CA.     Such  a  balance  is  tested  by  laying  a  certain  weight  on  different 
'parts  of  the   floor,  particularly  the  ends,  to   ascertain   whether  it 


INDEX  BALANCES.  H5 

everywhere  equilibrates  the  weight  G  £  times   smaller  than  itself 

6 
placed  on  the  scale. 

Remark  1.  Messrs.  George,  at  Paris,  manufacture  weighing-tables  of  a  peculiar  con- 
struction described  in  the  "  Bulletin  de  la  Societe  ^'Encouragement,  April,  1844.''  This 
balance  has  only  one  suspending  rod  BD,  Fig.  102  ;  but  to  provide  against  the  floor  FM 

Fig.  102. 


Burning,  there  are  two  knife-edge  axes  on  the  bank,  which  are  united  with  two  pair  of 
K'nife-edges  H  and  K.  by  four  parallel  rods  EH  and  FK.  According  to  the  theory  of 
couples,  the  tension  Q,  on  the  rod  BD  is  equal  to  a  weight  Q  laid  on  the  floor;  but  be- 
sides this,  the  floor  itself  acts  outwards  with  a  force  P  in  £,  and  with  an  opposite  force 
—  P  in  F  inwards.  If  d  be  the  distance  DLof  the  weight  Q  from  the  rod  BD,  and  e  the 
distance  of  the  knife  edges  E  and  F,  then  e  P  =  d  Q,  and,  therefore,  each  horizontal  force 

P  =  —  Q.    These  forces  do  not  influence  the  lever,  and  therefore  the  weight  Q  =  —  G, 

e  b 

if,  as  hitherto,  a  and  b  denote  the  lever  arms  CJl  and  CB.  and  G  the  weight  in  the  scale. 
Remark  2.  Weigh  bridges  are  treated  of  in  detail  in  the  "  Allgemeinen  Maschinen  En- 
cyclopadie,  Bd.  2,  Art.  Briickeuwaagen,"  under  the  art.  "Weighing  Machine,  in  the  En- 
cyclo.  Britannica  Edinensis." 

§  49.  Index  Balances,  or  Sent  Lever  Balances. — The  bent  lever 
balance  is  an  unequal-armed  lever  ACB,  Fig.  103,  which  shows  the 
weight  Q  of  a  body  hung  on  to  it  at  B,  by  an  index  CA  moving  over  a 
scale  DE,  the  weight  G  of  the  index  head  being  constant.  To  deter- 
mine the  theory  of  this  balance,  let  us  first  take  the  simple  case,  in 
which  the  axis  of  the  pointer  passes  through  the  point  of  suspension 
l?of  the  scale,  Fig.  104.  When  the  empty  balance  is  in  equilibrium, 
i.  e.  its  centre  of  gravity  S0  vertically  under  the  centre  of  motion  C, 
let  the  index  stand  in  the  position  CD0,  and  let  the  point  of  suspen- 
sion be  in  B0.  If  now  we  add  a  weight  Q,  then  B0  comes  to  B,  and 
JD0  to  D,  and  S0  to  S,  and  thus  the  weight  Q  acts  with  the  leverage 
CK,  and  the  weight  G  of  the  empty  balance,  with  the  leverage  CH. 
Therefore,  for  the  new  state  of  equilibrium  Q  .  CK  =  G  .  CH.  If 
now  D0N  falls  perpendicularly  on  CD,  we  have  CD0N  and  SCH, 

two  similar  triangles,  and,  therefore,  £5  =  -^-,  and  as  besides, 

Co 


116 


SPRING  BALANCES. 


C'K       T)  JV* 
the  triangles  D0PJY*and  CBK  are  similar,  we  have  also  7^=  -^p, 


Fig.  103. 


Fig.  104. 


and  hence: 

- 


,  CD=  CD 


d,  and  D0P  =  x,  Q=  -  .  -.   There- 
o      a 


fore  the  weight  Q  increases  with  the  portion  D0P  =  x,  cut  off  by  the 
index  from  the  vertical  D0L,  and  therefore  D0L  may  be  divided  as  a 
scale  of  equal  parts.  If  a  point  P  on  the  scale  has  been  found  for  a 
known  weight  put  on  the  balance,  other  points  of  division  are  got  by 
dividing  D0P  into  equal  parts.  If  the  index  centre  line  of  the  CD0 
does  not  pass  through  the  point  of  suspension,  but  has  another  di- 
rection CE0,  the  scale  of  equal  parts  corresponding  to  it,  is  found, 
if  we  place  the  right-angled  triangle  CD0L  as  CE0M  or  CE0,  and 
lastly,  in  order  to  get  the  circular  scale  E0R,  we  have  to  draw  radii 
from  the  centre  C  through  the  points  of  division  of  the  line  E0.M  to 
the  periphery  of  the  circle  passed  over  by  the  point  of  the  index. 

Remark.  There  are  other  index  balances  described  in  Lardners  and  Eater's  Mechanics. 
Such  balances  are  chiefly  used  for  weighing  letters  and  paper,  thread,  and  such  like 
manufactures,  where  samples  have  frequently  to  be  weighed. 

§  50.  Spring-balances  or  Dynamometers.  —  Spring-balances  are 
made  of  hardened  steel  springs,  upon  which  the  weights  or  forces 
act,  and  are  furnished  with  pointers,  indicating.  on  a  scale  the  force 
applied  in  deflecting  the  spring.  The  springs  must  be  perfectly 
tempered,  or  they  must  resume  their  original  form  on  removal  of  the 
force  applied.  Thus  spring-balances  should  never  be  strained  be- 
yond a  certain  point  proportional  to  their  strength  ;  for  if  we  surpass 
the  limits  of  their  elasticity  in  any  application  of  them,  they  are 


SPRING  BALANCES. 


117 


Fig.  105. 


afterwards  useless  as  accurate  measures  of  weight. 
The  springs  applied  for  such  balances  are  of  many 
different  forms.  Sometimes  they  are  wound  spirally 
on  cylinders,  and  enclosed  in  a  cylindrical  case,  so 
as  to  indicate  the  forces  applied  in  the  direction  of 
the  axes  of  the  cylinder  by  the  compression  or  ex- 
tension of  the  spiral.  In  other  balances  the  spring 
forms  an  open  ringJlBDEC,  Fig.  105,  and  the  index 
is  attached  by  a  hinge  to  the  end  C,  and  passed 
through  an  opening  in  the  end  A.  If  the  ring  B  be 
held  fast,  and  a  force  P  applied  at  E',  the  ends  A  and 
O  separate  in  the  direction  of  the  force  applied,  and 
the  index  CZ  rises  to  a  certain  position  on  the  scale 
fastened  at  D  to  the  spring.  If  the  scale  has  been 
previously  divided'  by  the  application  of  standard 
weights,  the  magnitude  of  any  force  P  applied,  though  previously 
unknown,  is  indicated  by  the  pointer. 

Fig.  106  is  a  representation  of  Regnier's  dynamometer.  ABCD 
is  a  steel  spring  forming  a  closed  ring,  which  may  either  be  drawn 
out  by  forces  P  and  P, 

or  pressed  together  by  FJg-  106- 

05  and  D;  DEGHis 
a  sector  connected  with 
the  spring,  on  which 
there  are  two  scales; 
MG  is  a  double  index 
turning  on  a  centre  at 
Jlf,  and  EOF  is  a  bent 
lever,  turning  on  0, 
and  which  is  acted  upon 
by  a  rod  .BE,  when  the 
parts  B  and  D  of  the 
spring  approach  each 
other  in  consequence 
of  the  application  of  weights  or  other  forces  as  above  mentioned. 
That  the  index  may  remain  where  the  force  has  put  it,  for  more  con- 
venient reading,  a  friction  leather  is  put  on  the  under  side. 

The  most  perfect,  and  most  easily  applied  dynamometer  for  me- 
chanical purposes,  is  that  described  by  Morin,  in  his  Treatise, 
"  Description  des  Appareils  chronome'triques  a  style,  et  des  Appareils 
dynamome'triques,  Metz,  1838,"  and  used  by  him  in  his  various  re- 
searches on  friction  and  other  important  mechanical  inquiries. 
Morin's  dynamometer  consists  of  two  equal  steel  springs  AB  and  CD, 
Fig.  107,  of  from  10  to  20  inches  in  length,  and  the  force  applied 
is  measured  by  the  separation  which  it  produces  between  the  two  spring 
plates  at  M.  In  order  to  determine  the  force,  for  example,  the  force 
of  traction  of  horses  on  a  carriage,  the  spring  plate  JV  is  connected 
by  a  bolt  with  the  carriage,  and  the  horses  are  attached  to  the  chain 


118 


SPRING  BALANCES. 


M  in  any  convenient  manner.     There  is  a  pointer  on  M  which  indi- 
cates on  a  scale  attached  to 

Fig.  107.  JV,  the  separation  of  the  plates 

produced  by  the  force  P  ap- 
plied. If  the  springs  be  plates 
of  uniform  breadth  and  thick- 
ness, and  be  I  =  the  length, 
b  =  the  breadth,  and  h  =  the 
thickness  ;  according  to  Vol. 
I.  §  190,  we  have  for  the 
deflexion  corresponding  to  a 
force  P  : 

PI3  PI3 


the  deflexion  increases  as  the 
force  applied,  and,  therefore, 
a  scale  of  equal  parts  should 
answer  in  this  dynamometer. 
As  the  deflexion  s  of  two 
springs  is  called  into  action, 
tlie  amount  is  double  of  that 
of  one  of  them,  or  it  is 

PI3 

and,    therefore, 


generally  s  =  -*_ ^  P,  if  $  be  a  number  determined  by  direct  experi- 
ment. If,  before  application  of  such  an  instrument,  a  known  weight 
or  force  be  put  upon  it,  and  the  deflexion  s  ascertained,  the  ratio 
between  force  and  deflexion  may  be  calculated,  and  a  scale  prepared. 
It  has  been  proved  by  experience  that  when  the  best  steel  is  used, 
the  deflexion  may  amount  to  -j1^  the  length,  without  surpassing  the 
limits  for  which  proportionality  between  force  and  deflexion  subsists. 
The  springs  that  have  been  employed  by  Morin  and  others  are  made 
into  the  form  of  beams  of  equal  resistance  throughout  their  length 
(Vol.  I.  §  204),  and  have,  therefore,  a  parabolic  form,  or  thicker  in 
the  centre  than  at  the  ends. 

Remark.  Forces  do  not  generally  act  uniformly,  but  are  continually  changing,  and 
therefore  the  usual  object  is  to  ascertain  the  mean  effort.  The  usual  index  dynamome- 
ters only  give  the  force  as  it  has  acted  at  some  particular  instant,  or  only  the  maximum 
effort.  There  is,  therefore,  extreme  uncertainty  in  the  indication  of  such  dynamometers, 
modified  as  they  have  been  by  M'Neill  and  others,  when  applied  to  measuring  the  effort 
of  horses  applied  to  ploughing,  canal  traction,  &c.  Morin  has  completely  provided  against 
this  defect,  by  attaching  to  his  dynamometer,  a  self  registering  apparatus,  first  suggested 
by  Poncelet,  (see  Morin's  work  above  quoted,)  by  which,  in  one  case,  the  force  for  each 
point  of  a  distance  passed  over  is  registered  in  the  form  of  a  curved  line,  drawn  on  paper, 
and  in  the  other  case,  the  force  as  applied  at  each  instant  is  summed  up  or  integrated 
by  a  machine  Both  apparatuses  give  the  product  of  the  force  into  the  distance  de- 
scribed, and,  therefore,  the  mean  effort  may  be  produced  when  the  mechanical  effect  is 
divided  by  distance  passed  over — by  a  canal  boat,  for  example. 

In  the  dynamometer  with  pencil  and  continuous  scroll  of  paper,  the  measure  of  the 
force  is  marked  by  a  pencil  pasting  through  M,  till  its  point  touches  a  scroll  of  paper 


FRICTION  BRAKE.  H9 

passing  under  it.  This  scroll  is  wound  from  the  roller  E,  (Fig.  107,)  to  the  roller  F 
which  is  set  in  motion  by  bands  or  wheel-work,  by  the  wheels  of  the  carriage  itself! 
When  no  force  acts  on  the  springs,  the  pencil  would  mark  a  straight  line  on  the  paper 
supposing  it  set  in  motion;  but  by  application  of  a  force  P,  the  springs  are  deflected,  and' 
therefore  a  line  more  or  less  tortuous  is  drawn  by  the  pencil  at  a  variable  distance 'from 
the  above  alluded  to  zero  line,  but  on  the  whole  parallel  to  it.  The  area  of  the  space 
between  the  two  lines  is  the  measure  of  the  mechanical  effect  developed  by  the  force  • 
for  the  basis  of  it  is  a  line  proportional  to  the  distance  passed  over,  and  the  height  is' 
itself  proportional  to  the  force  that  has  acted  to  bend  the 'spring. 

§  51.  Friction  Brake. — The  dynamometrical  brake  (Fr.  frein  dy- 
namometrique  de  M.  Prony],  is  used  to  measure  the  power  applied 
to,  and  mechanical  effect  produced  by  a  revolving  shaft,  or  other 
revolving  part  of  a  machine.  In  its  simplest  form,  this  instrument 
consists  of  a  beam  JIB,  Fig.  108,  with  a  balance  scale  AG\  and  of 
two  wooden  segments  D  and  EF,  which  can  be  tightened  on  the 
revolving  axis  C,  by  means  of  screw-bolts  EH  and  FK.  To  mea- 
sure, by  means  of  this  arrangement,  the  power  of  the  axis  C  for  a 
given  number  of  revolutions,  weights  are  laid  in  the  scale,  and  the 
screw-bolts  drawn  up  until 

the  shaft  makes  the  given  Fig.  108. 

number  of  revolutions,  and 
the  beam  maintains  a  ho- 
rizontal position,  without 
support  or  check  from  the 
blocks  L  or  R.  In  these 
circumstances  the  whole 
mechanical  effect  expend- 
e.d  is  consumed  in  over- 
coming the  friction  between  the  shaft  and  the  wooden  segments,  and 
this  mechanical  effect  is  equal  to  the  work  or  useful  effect  of  the 
revolving  shaft.  As,  again,  the  beam  hangs  freely,  it  is  only  the 
friction  F  acting  in  the  direction  of  the  revolution  that  counter- 
balances the  weight  at  G,  and  this  friction  may  be  deduced  from  the 
weights.  If  we  put  the  lever  CM  of  the  weight  G  referred  to  the 
axis  of  the  shaft  =  a,  the  statical  moment  of  the  weight,  and  there- 
fore also  the  moment  of  the  friction,  or  the  friction  itself  acting  with 
the  lever  equal  to  unity  =  G  a,  then,  if  t  represent  the  angular  ve- 
locity of  the  shaft,  the  mechanical  effect  produced  L  =  Pv  =  Ga  .  e 
=  f  a  G  per  second. 

If,  again,  u  =  the  number  of  revolutions  of  the  shaft  per  minute, 

then  f  =  l^f  =  ?2,  and,  therefore,  the  work  required  L  =  I^L  G. 
60        30  <>u 

The  weight  G  must  of  course  include,  not  only  the  weight  in  the 
scales,  but  the  weight  of  the  apparatus  reduced  to  the  point  of  sus- 
pension. To  do  this,  the  apparatus  is  placed  upon  a  knife-edge  at 
.D,  and  a  cord  from  A  attached  to  a  balance  would  give  the  weight 
required. 

The  friction  brake  as  represented  in  Fig.  109,  with  a  cast  iron 
friction  ring  DEF  is  a  convenient  form  of  this  instrument.  This 
ring  is  fastened  by  three  pairs  of  screws  on  any  sized  shaft  that  will 


120 


FRICTION  BRAKE. 


pass  through  the  ring.     For  the  wooden  segment  an  iron  band  is 
substituted,  embracing  half  the  circum- 
Fis- 109-  ference  of  the  iron  ring.      The  band 

ends  in  two  bolts  passing  through  the 
beam  AB,  and  may  be  tightened  at  will 
by  means  of  screw  nuts  at  H  and  K. 

To  hinder  the  firing  of  the  wood,  or 
excessive  heating  of  the  iron,  water  is 
continually  supplied  through  a  small 
hole  L.  This  apparatus  is  known  in 
Germany  as  "Egen's  Friction  Brake." 

Example.  To  determine  the  mechanical  effect  produced  by  a  water  wheel,  a  friction 
brake  was  placed  on  the  shaft,  and  when  the  water  let  on  had  been  perfectly  regulated 
for  six  revolutions  per  minute,  the  weight  G  including  the  reduced  weight  of  the  instru- 
ment was  530  Ibs.,  the  leverage  of  this  weight  was  a  s=  10,5  feet.  From  these  quanti- 
ties we  deduce  the  effect  given  off  by  the  water  wheel  to  have  been : 
Z  _  «•  •  6  •  10<5  .  530  _  3497  a  lbg-  _  ^  horse  power. 

§  52.  In  more  recent  cases,  various  forms  of  friction  brake  have 
been  adopted,  some  of  them  very  complicated.  The  simplest  we 

know  of  is  that  of  Armstrong, 

Fig.  no.  shown  in  Fig.  110.    This  con- 

sists of  an  iron  ring,  which  is 
tightened  round  the  shaft  by 
a  screw  at  B,  and  of  a  lever 
ADE  with  a  scale  for  weight 
G  on  one  side,  and  a  fork* 
shaped  piece  at  the  other, 
which  fits  into  snuggs  project- 
ing from  the  ring.  There  is  a 
prolongation  of  one  prong  of  the  fork,  by  which  the  weight  of  the 
instrument  itself  can  be  counterbalanced,  and  which  is  otherwise  con- 
venient in  the  application  of  the  instrument. 

Navier  proposed  a  mode  of  determining  the  effect  given  off  at  the 
circumference  of  a  shaft  by  laying  an  iron  band  round  the  shaft, 
attaching  the  one  end  of  this  to  a  spring  balance,  the  other  end  being 
weighted,  so  that  the  friction  on  the  wheel  causes  a  resistance,  in 
overcoming  which  only  the  required  number  of  revolutions  take 
place.  The  difference  of  this  weight  Q  and  that  indicated  by  the 
spring  balance  P,  is  of  course  =  the  friction  F  between  the  shaft 
and  the  band.  If  then  p  be  the  circumference  of  the  shaft,  and  n 
the  number  of  revolutions  per  minute,  the  effect  produced 


When  a  spring  balance  cannot  be  obtained,  a  simple  band,  as 
shown  in  Fig.  Ill,  is  sufficient  for  the  purpose,  if  the  experiment  be 
made  twice,  and  the  end  B  be  fastened  to  an  upright  or  other  fixture, 
first  on  the  one  side  and  then  on  the  other  of  the  shaft.  In  this  way 
one  experiment  gives  us  Q  =  P  +  F,  and  in  the  other  Q  =  P, 


THE  POWER  OF  ANIMALS. 


121 


because  in  the  one  case,  the  friction  F  acting  in  the  direction  of 
the  revolution  of  the  shaft, 

counteracts     the     weight  Fig.  in. 

hanging  in  the  scale  on 
the  end  rf,  and  in  the  other 
it  acts  with  this  friction. 
For  this  arrangement,  used 
by  the  author  in  many  ex- 
periments, the  mode  of  cal- 
culation already  explained 
applies  precisely.  As  the 
power  has  only  a  small 
leverage  in  this  arrange- 
ment, it  is  only  suitable  for 
cases  in  which  the  effort 
exerted  is  small.  The  strain  on  the  band  may  be  multiplied  by  means 
of  an  unequal-armed  lever  attached  at  A,  instead  of  the  direct  appli- 
cation of  the  weight  in  the  scale.  The  author  has  successfully  applied 
a  leverage  of  10  to  1  in  this  way.  In  order  to  avoid  the  objectionable 
increase  of  friction  of  the  axle  or  gudgeons  induced  by  this  appa- 
ratus, the  band  may  be  made  to  pass  round  the  shaft,  carrying  the 
one  end  upwards  and  the  other  down. 

Remark.  Egen  treats  of  the  different  forms  of  dynamometers  in  his  work,  entitled 
"  Untersuchungen  iiber  den  Eflect  einiger  Wasserwerke,  &c.,"  and  Hiilsse,  in  article 
Bremsdynamometer  in  the  "  Allgemeinen  Maschinenencyclopadie.v  James  White,  of 
Manchester,  invented  the  friction  brake  in  1808.  See  Hachette  "  Traite  elementaire  des 
Machines,  p.  460."  Prony's  original  paper  is  in  the  "Annales  des  Mines,  1826.  There 
are  remarks  on  its  use  in  the  writings  of  Poncelet  and  Morin,  worthy  attention.  W.  G. 
Armstrong's  paper  is  in  the  "Mechanic's  Magazine,"  vol.  xxxii.  p.  531.  Weisbach's 
papers  on  the  friction  band,  in  the  "  Polytechnishes  Central  Blatt,''  1844. 


CHAPTER   II. 

OF  ANIMAL  POWER,  AND  ITS  RECIPIENT  MACHINES. 

§  53.  The  Power  of  Animals.— The  working  power  of  animals  is, 
of  course,  not  only  different  for  individuals  of  different  species,  but 
for  animals  of  the  same  species.  The  work  done  by  animals  of  the 
same  species  depends  on  their  race,  age,  temper,  and  management, 
as  well  as  on  the  food  they  get,  and  their  keeping  generally,  and 
also  on  the  nature  of  the  work  to  which  they  are  applied,  or  the 
manner  of  putting  them  to  their  work,  &c.  We  cannot  discuss  these 
different  points  here,  but  for  each  kind  of  animal  employed  by  man, 
we  shall  assume  as  fair  an  average  specimen  as  possible, — that  the 
animal  is  judiciously  applied  to  work  it  has  been  used  to  perform, 
and  that  its  food  is  suitable.  But  the  working  capabilities  of  ani- 
mals depend  also  on  the  effort  they  exert,  and  on  their  speed,  and 
VOL.  II. — 11 


122  THE  POWER  OF  ANIMALS. 

on  the  time  during  which  they  continue  to  work.  There  is  a  certain 
mean  effort,  speed,  and  length  of  shift  for  which  the  work  done  is  a 
maximum.  The  greater  the  effort  an  animal  has  to  exert,  the  less 
the  velocity  with  which  it  can  move,  and  vice  versd.  And  there  is  a 
maximum  effort  when  the  speed  is  reduced  to  nothing,  and  when, 
therefore,  the  work  done  is  nothing.  It  is  evident,  therefore,  that 
animal  powers  should  work  with  only  a  certain  velocity,  exerting  a 
certain  mean  effort,  in  order  to  get  the  maximum  effect;  and  it  also 
appears  that  a  certain  mean  length  of  shift,  or  of  day's  work,  is  ne- 
cessary to  the  same  end.  Small  deviations  from  the  circumstances 
corresponding  to  the  maximum  effect  produced,  are  proved  by  long 
experience  to  be  of  little  consequence.  It  is  also  a  matter  of  fact 
that  animals  produce  a  greater  effect,  when  they  work  with  variable 
efforts  and  velocities,  than  when  these  are  constant  for  the  day. 
Also  pauses  in  the  work,  for  breathing  times,  makes  the  accomplish- 
ment of  the  same  amount  of  work  less  fatiguing,  or  the  more  the 
work  actually  done  in  a  unit  of  time  differs  from  the  mean  amount 
of  work,  the  less  is  the  fatigue. 

The  main  point  to  be  attended  to  in  respect  to  animal  powers  is 
the  "  day's  work,"  If  this  be  compared  with  the  daily  cost  of  main- 
taining the  animals,  and  interest  on  capital  invested,  we  have  a  mea- 
sure of  the  value  of  different  animal  powers. 

§  54.  The  manner  and  means  of  employing  the  power  of  men  and 
animals  is  very  different.  Animal  powers  produce  their  effects  either 
with  or  without  the  intervention  of  machines.  For  the  different 
means  of  employing  labor,  the  degree  of  fatigue  induced  is  not  pro- 
portional to  the  work  done.  Many  operations  fatigue  more  than 
others;  or  what  amounts  to  the  same  thing,  the  mechanical  effect 
produced  is  much  smaller  in  some  modes  of  applying  labor  than  in 
others.  Again,  all  labor  cannot  be  measured  by  the  same  standard 
as  is  involved  in  our  definition  of  mechanical  effect.  The  work  done 
in  the  transport  of  burdens  on  a  horizontal  road  cannot  be  referred 
to  the  same  standard  as  the  raising  of  a  weight  is  referred  to.  Ac- 
cording to  the  notions  we  have  acquired  hitherto,  the  mechanical 
effect  produced  in  the  transport  of  burdens  on  a  horizontal  road  is 
nothing,  because  there  is  no  space  described  in  the  direction  of  the 
force  (Vol.  I.  §  80)  exerted,  that  is,  at  right  angles  to  the  road; 
whilst  in  drawing  or  lifting  up  a  weight,  the  work  done,  or  mechani- 
cal effect  produced,  is  determined  by  the  product  of  the  weight  into 
the  distance  through  which  it  has  been  raised.  It  is  true,  that 
walking  or  carrying  fatigues  as  much  as  lifting  does,  i.  e.  the  "day's 
work"  is  consumed  by  the  one  as  by  the  .other  kind  of  labor;  and, 
therefore,  a  certain  day's  work  is  attributable  to  the  one  as  there  is 
to  the  other,  although  they  are  essentially  different  in  their  nature. 
According  to  experience,  a  man  can  walk,  unburdened,  for  ten 
hours  a  day  at  4£  feet  per  second  (something  under  3 J  miles  per 
hour).  If  we  assume  his  weight  at  140  Ibs.  we  get  as  the  day's 
labor  140  .  4,75  .  10  .  60  .  60  =  23,940000  ft.  Ibs. 

If  a  man  carry  a  load  of  85|  Ibs.  on  his  shoulders,  he  can  walk 


FORMULAS.  123 

for  7  hours  dAily  with  a  speed  of  2,4  feet  per  second,  and,  there- 
fore, produces  daily  the  quantity  of  work  ==  85,5  .  2,4  .  7  .  60  .  60 
=  5',171000#.  Ibs.,  neglecting  his  own  weight. 

A  horse  will  carry  256  Ibs.  for  10  hours  daily,  walking  3£  feet 
per  second,  so  that  its  day's  work  amounts  to  256  .  3,5  .  10  .  60  . 
60  =  32'256000  ft.  Ibs.,  or  more  than  6  times  as  much  as  a  man 
doing  the  same  kind  of  work.  If  the  horse  carries  only  171  Ibs.  on 
his  back,  he  will  trot  at  7  feet  per  second  for  7  hours  daily,  and  the 
work  done  in  this  case  is  only  171  .  7  .  7  .  60  .  60  =  30'164400  ft. 
Ibs.  daily. 

The  amounts  of  work  done  in  raising  burdens  is  much  smaller, 
for  in  this  case  mechanical  effect,  according  to  our  definition,  is 
produced,  or  the,  space  is  described  in  the  direction  of  the  effort 
exerted. 

If  a  man,  unburdened,  ascend  a  flight  of  steps,  then,  for  a  day's 
work  of  8  hours,  the  velocity  measured  in  the  vertical  direction  is 
0,48  feet  per  second  ;  therefore,  the  amount  of  work  done  daily 
=  140  .  0,48  .  8  .  60  .  60  =  1'935000  ft.  Ibs.  It  thus  appears  that 
a  man  can  go  over  12  J  times  the  space  horizontally  that  he  can  ver- 
tically. 

In  constructing  a  reservoir  dam,  the  author  observed 
that  4  practised  men,  raised  a  dolly,  Fig.  112,  weigh-       Fi&-  us- 
ing 120  Ibs.,  4  feet  high  34  times  per  minute,  and  after 
a  spell  of  260  seconds,  rested  260  seconds;  so  that,  on    < 
the  whole,  there  were  only  5  hours  work  in  the  day. 
From  this  it  appears  that  the  day's  work  of  a  man 

=  1?2.  .  4  .  34  .  5  .  60=  1'224000  feet  Ibs. 
4 

Remark  1.  In  the  "  Ingenieur,"  there  is  detailed  information  on  the  work  done  by  ani- 
mal power.  In  the  sequel,  the  effect  produced  by  animals  by  aid  of  machines  is  given 
for  each  machine  respectively. 

Remark  2.  The  effect  produced  by  men  and  animals  is  far  from  being  accurately 
ascertained.  The  effect  produced  by  men  working  under  disadvantageous  circumstances, 
or  by  unpracticed  laborers,  is  not  one-half  of  that  produced  by  well-trained  hands.  Cou- 
lomb, in  his  "  Theorie  des  Machines  simples,"  first  entered  on  investigations  of  the  effect 
of  animal  powers.  Desaguiliers  ("Cours  de  Physique  experimentale,")  and  Schulze 
("  Abhandlungen  der  Berliner  Akademie,")  had  previously  occupied  themselves  with 
the  subject.  Many  experiments  have  been  made  and  recorded  in  more  recent  times. 
See  Hachette,  "Traite  elementaire,  &c.,"  Morin,  "Aide  Memoire,"  Mr.  Field  in  the 
"Transactions  of  the  Institution  of  Civil  Engineers,  London,"  Sim's  "  Practical  Tunnel- 
ing," and  Gerstner's  "Mechanik,"  Band  1. 

§  55.  Formulas.  —  Effort  and  velocity  have  a  very  close  depend- 
ence in  the  application  of  animal  power;  but  the  law  of  their  de- 
pendence is  by  no  means  known,  and  is  still  less  deducible  d  priori. 
The  following  empirical  formulas,  given  by  Euler  and  Bouguer,  are 
only  to  be  considered  as  approximations.  If  Kl  be  the  maximum 
effort  which  an  animal  can  exert  without  velocity,  and  ev  the  greatest 
velocity  it  can  give  itself  when  unimpeded  by  the  necessity  of  ex- 
traneous effort,  we  have  for  any  other  velocity  and  effort  : 


according  to  Bouguer  :  P  =•  M  --  \ 

\  <?j/ 


124 


FORMULAS. 


ac3ording  to  Euler  :  P 


—  —  8) 


Euler  :P=l_ 


The  first  of  these  is  the  most  simple,  and  that  which,  according  to 
Gerstner,  corresponds  best  with  observation.  According  to  Schulze's 
observations,  on  the  other  hand,  the  last  formula  appears  to  be  most 
consistent  with  experiment.  If  we  draw  v  as  abscissa,  and  P  as 
ordinates  to  a  curve,  the  first  formula  corresponds  to  a  straight  line 
JIB,  Fig.  113,  the  second  with  a  concave  parabolic  curve  ^P2S,  and 
the  third  with  a  convex  parabolic  curve  *flP3B,  and  the  ordinates 
MPV  of  the  straight  line  always  lie  between  the  ordinates  MP2  and 
MP3  of  the  two  parabolic  curves.  The  abscissa  OM,  for  example, 
=  v  =  %  cl  corresponds  to  the  ordinates  MPl  =  J  K.  =  £  (X/3,  also 
MP2=$  K=$  OA,  and  MP^=\  tf=  \OA. 
The  formula  of  Bouguer,  therefore,  gives  values 
of  the  effort  which  lie  between  the  values  given 
by  the  two  formulas  of  Euler  ;  and  we  may, 
therefore,  make  use  of  Bouguer's  formula  until 
some  special  reason  for  adopting  Euler's  for- 
mula be  adduced.  If  we  introduce  into  Bou- 
guer's formula,  instead  of  the  maximum  values 
Kl  and  Cj,  the  halves  of  these,  or  their  mean 
values  K=  %  Kv  and  c  =  %  cv  we  get  a  formula  first  applied  by 
Gerstner  : 


Fig.  1  1  3. 


and  inversely,  v 


Although  this  formula  can  be  but 


/     P\ 

(2  --  )< 

little  depended  upon  as  accurate  for  extreme  values  of  v  and  P,  yet 
it  may  be  presumed,  that  for  values  not  very  different  from  the 
mean,  they  are  sufficiently  near  for  practical  uses.  The  mechanical 
effect  produced  per  second  would  follow  from  this  : 

Pv  =  (z  —  -\vK.    As(2  —  -\vK=(2c—v)v-, 

the  mechanical  effect  is  a  maximum,  as  in  Vol.  I.  §  386,  when  v  =  c, 
or  when  P  =  K,  or  when  the  velocity  and  effort  are  mean  values, 
i.  e.,  Pv  =  Kc.  If  we  try  to  get  a  greater  or  less  velocity,  or  a 
greater  or  less  effort,  we  get  an  effect  L  =  Pv  less 
than  Kc.  If  we  set  off  the  velocities  as  abscisses, 
and  the  amounts  of  mechanical  effect  produced  as 
ordinates,  we  get  as  the  projected  curve  a  para- 
bola ADB,  Fig.  114  ;  and  it  is  evident  that  not 
only  for  abscissa  JIM  <  J1C,  but  also  for  AMl  > 
BC,  the  ordinates  MP,  MjPv  are  less  than  for 

the  abscissa  JJC=c.  For  v  =  -  ,  as  also  for  v  =  | 
c,  it  follows  from  the  above  that  L  =  f  Kc  =  f  CD. 


Fig.  114. 


WOKK  DOXE  BY  AID  OF  MACHINES. 


125 


According  to  Gerstner,  the  following  table  represents  the  drauaht 
of  animals  applied  properly  to  draw  by  traces. 


Animals. 

Weight. 

Mean  ef- 
fort K  in 
Ibs. 

Mean  speed  I  Mean  period 
c  in  feet  per  of  day's  work, 
second.              Hours. 

Effect  pro- 
duced p.  sec. 
in  ft.  Ibs. 

Daily  effect 
feet  Ibs. 

Man      . 
Horse    . 
Ox    .     . 

150 
600 
600 

30 
120 
120 

2,5 
4 
2,5 

8 
8 
8 

75 
480 
300 

2'  160000 
13'824000 
8'640000 

Ass  .     . 

360 

72 

2,5 

8 

180 

5'  184000 

Mule     . 

500 

100 

3,5 

8 

350 

IQ'080000 

Exampk  1.  According  to  the  above  table,  a  man  working  with  an  effort  of  30  Ibs., 
and  mean  velocity  of  2$  feet  per  second,  produces  in  a  day  an  amount  of  mechanical 
effect  represented  by  2'  160000  feet  Ibs.  If  he  be  urged  to  work  at  3  feet  per  second, 

the  effort  will  be  reduced  to  P=  (2  —  A-^  30  =  24  Ibs.,  and  his  daily  effect  would 

only  be  24  .  3  .  8  .  60  .  60  =2'073600  feet  Ibs. 

Exampk  2.  If  a  horse  be  obliged  to  draw  with  an  effort  of  150  Ibs.,  it  can  only  be 
done  with  a  velocity  v  =  /  3  --  J  4  =  3  feet  per  second,  and  thus  his  effective 

work  is  reduced  to  only  3  .  150  =  450  feet  Ibs.  per  second. 

Remark.  Fourier,  in  the  dnnales  des  Fonts  et  Chaussees,  1836,  gives  a  complicated  for- 
mula for  the  effect  produced  by  horses.  See  also  Crelle's  Journal  der  BauJcunst  Band 
xii.  1838. 

§  56.  Work  done  by  aid  of  Machines.  —  If  we  follow  Gerstner's 
notion,  that  the  period  or  time  of  each  shift,  or  day's  work,  has  the 
same  influence  on  the  amount  of  work  done  as  the  velocity,  we  must 
then  put  for  the  effort: 


and  from  this  we  get  the  daily  effect  produced: 


There  can  be  no  doubt  that  the  effect  produced  is  a  maximum, 
that  is  =  K  c  t,  when  the  animal  is  made  to  work,  not  only  with  a 
mean  velocity  and  effort,  but  also  when  the  time  of  work  is  kept 
within  the  mean  for  this.  It  is  to  be  kept  in  mind,  however,  that 
this  formula  only  applies  when  the  values  of  u,  z,  and  P  do  not 
differ  widely  from  c,  £,  and  K. 

M.  Maschek,  of  Prague,  recommends  the  expression: 


which  is  certainly  more  convenient  for  calculation.* 

Eight  to  ten  hours  per  day  is  a  good  average  day's  work,  and, 

therefore,  the  factor  (%  —  -)  may  generally  be  neglected,  or  the 
day's  effect  may  be  written  £=  (2  —  -\  K  v  z.      If,  however,  an 


*  Neue  Theorie  der  menschlichen  und  thierischen  Krifte,  &c.,  von  F.  J.  Maschek,  Prag. 
11* 


126  WORK  DONE  BY  AID  OF  MACHINES. 

animal  be  applied  to  a  machine,  its  effort  P  would  be  divided  into 
an  effort  P1  for  doing  the  work,  and  an  effort  P2  for  overcoming  pre- 
judicial resistances,  or  P  =  Pl  +  P2,  both  resistances  being  reduced  to 
the  point  of  application  of  the  effort.  It  is  also  usual,  as  we  shall 
learn  in  the  sequel,  to  find  the  prejudicial  resistances  P2  composed 
of  a  constant  part  jR,  independent  of  the  strain  on  the  machine,  and 
a  part  &  .  P^  proportional,  or  nearly  so,  to  the  useful  effect  produced 
or  work  done,  where  8  is  co-efficient  derived  from  experiment,  thus 
P2  =  R  +  s  .  Pjj  and,  therefore, 

P=(l+«)  P^R;  and  again  (2  —  -)  K=  (l  +  «)  P!  +  R- 
The  total  effect  produced  per  second  is,  therefore, 


Pv  =  /2  —  -)  Kv=  (1  +  a) 


and,  therefore,  the  useful  effect  produced: 
(2K—  R)v—— 


-         c-v 


~\v.       K 

J     i+« 


That  this  effect  may  be  the  greatest  possible  (see  previous  para- 
graph), we  must  have  v=i/2  --  \c—  t\  --  \c,  or  the  velo- 

city less  than  the  mean  velocity;  and  so  much  the  less,  the  greater 
the  constant  part  of  the  prejudicial  resistance  is.  The  effort  corre- 
sponding would  be,  according  to  this  : 


or  greater  than  the  mean  effort.      The  useful  resistance,  on  the 

*-f 

other  hand,  is  P,  =  —  —  —  .     The  total  effect  produced  is: 
1  +  a 

Pv  =  |l  —  (-9-^)  1  %  c)  and  tne  useful  effect  produced  is: 

Pj>  =  (  1  —  o-rr  )  =  —  —  ,  and  the  efficiency  of  the  machine  : 
\  ZA  /  1  -f-  8 


Example.  If  in  a  machine  turned  by  two  horses,  the  constant  prejudioal  resistance  re- 
duced to  the  point  of  application  of  the  horses'  effbrt=60  Ibs.,  the  velocity  at  which 
the  horses  should  work  when  K  =  2  .  120  ==240  Ibs.,  and  c  =  4  feet,  is  reduced  to  v 
=  (].  —  J^)  c  =  I  .  4  =  3,5  feet.  Further,  the  effort  of  the  horses  =  240  +  — 

=270  Ibs.,  and.  therefore,  that  of  one  horse  =  135  Ibs.  If,  now,  the  constant  part  of 
prejudicial  resistance  be  15  per  cent,  of  the  useful  resistance,  then  J=0,15,  and,  there- 
fore, the  resistance  to  be  put  on  the  machine  P.  — 240~30  =  182,5  Ibs.,  and  the  effi- 

1,15 
ciency  of  the  machine  would  be  i,  =  (|)3-j-  1,15  =  0,67. 


THE  LEVER.  J27 

Remark.  Gerstner  reduces  the  calculation  of  the  effect  of  animal  power  to  motion  on 
an  inclined  plane.  If  G  be  the  weight  of  the  animal,  P  the  effort  exerted,  and  tt  the 
angle  of  inclination  of  the  inclined  plane,  upon  which  the  moving  power  ascends  with 
its  load,  then  the  effort  is  P  +  G  sin.  .  (see  Theory  of  Inclined  Plane,  Vol.  L  §  134),  and 
hence  (2  —  l)  K=  P+  G  sin.  a.  Hence,  we  have  the  load  with  which  an  animal 
can  ascend  an  inclined  plane,  and,  conversely,  the  inclination  corresponding  to  a  given 


load,  viz.:  sin.  a, 


,  thus  when  P  =  0,  and  v  =  c,  or  when  the  animal 


Fig.  115. 


has  no  resistance  to  overcome,  and  goes  with  the  mean  velocity  sin.  a.  =  _.     But,  the 

weight  of  an  animal  is  almost  always  five  times  as  great  as  the  mean  effort  it  can  exert, 
therefore  sin.  «=}  and  «=  11^°  is  the  angle  of  inclination  of  a  plane  which  an  animal 
can  ascend  with  the  mean  amount  of  exertion  and  fatigue.  This  corresponds  to  a  rise 
of  one  foot  in  five  feet,  or  nearly  so. 

§  57.  The  Lever.  —  Animal  powers  are  applied  to  work  by  means 
of  the  lever  or  the  wheel  and  axle.  The  latter  are  either  horizontal 
or  vertical.  We  shall  first  speak  of 
the  lever  as  a  machine  for  receiving 
(and  transmitting)  animal  power.  The 
general  theory  of  this  machine  is  known 
from  Vol.  I.  §§  126,  127,  and  170. 
The  lever  is  either  single  as  J1CB,  Fig. 
115,  or  double,  as  ACBAV  Fig.  116  ;  the 
one  has  only  one  arm  for  the  applica- 
tion of  the  power  C/f,  whilst  the  other 
has  two  arms  CA  and  C^.  The  lever 
produces  an  oscillating  circular  motion,  and  is,  therefore,  chiefly 
applied,  when  a  reciprocating  up  and  down  motion  is  desired,  as  in 

Fig.  116. 


pumping.  Handles,  suited  to  the  number  of  hands  to  be  applied, 
are  affixed  to  the  lever.  As  the  strength  can  be  better  exerted  in 
pulling  downwards  than  in  lifting  upwards,  it  is  usual  to  make  the 
down-stroke  the  working-stroke,  and  counter-balances  are  attached  so 
as  to  aid  the  workmen  in  the  up-8troJce,  or  the  double  lever  is  used, 
on  which  the  workers  alternately  pull  downwards.  When  the  down- 
stroke  is  the  effective  stroke,  ropes,  hanging  from  the  end  of  the 
lever,  are  frequently  substituted  for  handles.  Levers  are  sometimes 
moved  by  the  tread  of  the  feet. 

That  there  may  not  be  too  great  a  change  of  direction  during  a 
stroke,  the  lever's  motion  is  confined  to  an  arc  of  not  more  than 


128 


THE  LEVER. 


60°,  and,  in  order  to  facilitate  the  exertions  of  the  power,  the  space 
passed  through  at  each  stroke  is  kept  proportional  to  the  length  of 
arm  of  the  workers,  or  at  from  2|  to  3J  feet.  Again,  the  handles 
should  not  come  within  from  3  to  3£  feet  from  the  floor.  According 
to  experience,  men  work  8  hours  per  day,  exerting  an  effort  of  k 
=  10,7  Ibs.  on  the  end  of  a  lever,  with  a  velocity  c  =  3,5  feet. 
Therefore,  the  mechanical  effect  produced  by  a  man  applied  to  a 
lever,  as  in  pumping,  is  per  second:  L  =  10,7  .  3,5=  37,45  ft.  Ibs., 
and,  therefore,  the  day's  work 

=  Kct=  37,45  .  8  .  3600  =  1'078560  ft.  Ibs. 

In  putting  up  a  lever,  it  is  necessary  to  take  care  that  the  work- 
men shall  be  applied  so  as  to  exert  the  ascertained  mean  effort  with 
the  mean  velocity;  or  rather,  that  the  effective  effort  shall  exceed 
the  mean  effort  by  only  one-half  of  the  constant  prejudicial  resistance. 

The  lever  itself  is  subject  to  only  one  prejudicial  resistance,  viz. : 
the  friction  of  the  fulcrum.  If  R  be  the  pressure  on  the  fulcrum 
arising  from  the  weight  of  the  lever  and  from  the  effort  and  resist- 
ance, r  the  radius  of  the  fulcrum,  f  the  co-efficient  of  friction,  and  a 
the  leverage  CA  of  the  power,  then  the  axle  friction  reduced  to  the 

point  of  application  of  the  power  F=J—R;  as,  however,  f,  and  also 

a 

-  are  generally  small  fractions,  F  is  so  small  that  it  may  be  neglect- 
a 

ed  in  most  cases,  compared  with  the  other  resistances. 

If  we  suppose  a  useful  resistance  Q  and  a  prejudicial  resistance 
S  Q  -f  W  acting  at  the  point  jB,  and  if  we  put  the  leverage  CB  of 
these  resistances  =  6,  the  moment  of  the  effort  becomes: 
Pa  =  [(!  +  «)  Q+  W]  b,  and,  therefore,  the  effort  itself: 

P  =  -  [(!  +  «)  Q+JF].     But,  that  the  power  of  men  may  be  most 

advantageously  applied :   P  =  K  +    -  .   —  and,  therefore,  -  K 

a         2  b 

W 

=  (1  +  «)  Q  4-  -jp  and,  therefore,  the  ratio  of  the  lever  arms 

a        (1  +  MQ  +  ifF.    .    , 

-  =  ^ — ' —    „. —  is  to  be  employed. 
b  K 

Remark.  The  arms  of  the  lever  are  varia- 
ble to  a  certain  extent  during  the  stroke,  and. 
therefore,  it  may  be  well  to  determine  the 
amount  of  this  variation. 

If  the  arm  CB,  Fig.  117,  be  horizontal  at 
the  half-stroke,  and  if  the  angle  BtCB2  passed 
through  in  a  stroke  =  8°,  the  height  through 
which  the  resistance  is  overcome  s  =  BtBt 
=  2  6  sin.  — ,  and,  therefore,  the  mechanical 
effect  produced  or  expended  in  one  stroke 
=  2  bsin.  -  .  Q.  If,  however,  the  resist- 
ance were  constant  during  the  stroke  at  a 


WINDLASS.  129 

leverage  CB  =  b,  the  space  passed  over  at  each  stroke  would  be  =  arc  BtBB2  =  56,  and, 

2  b  sin.  B- 
therefore,  the  resistance  would  be  =:____?  Q__2  "M  *  .  Q)  and  the  statical  mo. 

ment  =  -  '-±—  Q  b.     Conversely,  we  may  assume  that  the  resistance  Q  acts  during 

a  stroke  on  the  mean  length  of  arm         *''"'  *        Forfi0  =  60°  this  lever  =        ^ 

B  arc,  60° 

=  -  -  =0,955  b,  or  not  quite  5  per  cent  less  than  6,  and  for  smaller  arcs  of  oscilla- 

tion, the  difference  is  still  much  less. 

Exampk.  What  proportion  of  arms  should  be  chosen  for  a  lever,  that  for  a  useful  re- 
sistance of  160  Ibs.  and  a    prejudicial   resistance  Q,  =  0,15  Q-j-  55  =  0,15  .  160-|-  55 
=  79  Ibs.,  four  men  may  work  to  the  best  advantage?     K=  4  .  10,7^42,8  Ibs.  there- 
-        a        1,15.1604-155       2115       .„      lr   , 
fore  _  =  _  -  L_±  -  —  -  =  4,9.     If  the  resistance  passes  through  1  foot  for 

b  42,8  428 

each  stroke,  the  power  must  at  the  same  time  pass  through  4,9  feet,  and  if  we  take 

the  angle  of  oscillation  |S^500,  we  get  for  the  suitable  length  of  lever  ft  •  —  _ 


—  _  °'5     =  1,183  feet,  and  the  length  of  arm  a  =  4,9  .  6  =  4,9  .  1.183  =  5,80  ft.     The 
sin.  25° 

effort  necessary  is  P  =  160+  79  =  48,78    Ibs.,   therefore,  the  effort  of  each   man  = 
12,1  95  Ibs.,  and  the  efficiency 

„  =  (  1  --  ™  _  Y=  (!—  Q--1311)'  __  o,657.     We  see,  therefore,  that  four  men 
\         2  .  4,9  .  42,87  1,15 

1~75 

capable  of  a  day's  work  each  =  1'075S60  ft.  Ibs.,  or  4'303440  ft.  Ibs.  in  all,  would  only 
produce  0,657  .  4'303440  =  2'800000  feet  Ibs.  useful  effect  with  this  machine. 

§  58.    Windlass.  —  The   best  means  of  applying  the  power  of 
men,  is  the  windlass  (Fr.  treuil,  tour;  Ger.  ffaspel).     This  machine 
consists  of  a  horizontal  axle,  at  the  circumference  of  which  the  re- 
sistance acts,  and  of  a  crank,  handle,  or 
winch,  Fig.  118,  or  series  of  handles  on  Fi£-  118- 

a  wheel,  Fig.  120,  or  of  fixed  or  mova- 
ble levers  (hand-spikes),  Fig.  119.  With 
the  winch,  the  laborers  have  a  continu- 
ous hold  throughout  the  revolution, 
whilst  with  the  wheel  or  hand-spike,  the 
action  is  hand  over  hand,  or  otherwise 
at  short  intervals.  The  winch  is  the 
form  used  for  general  purposes.  The 
wheel  is  applied  principally  in  working 

the  tiller  on  board-ship,  and  the  movable  levers  are  chiefly  used  for 
weighing  anchor  by  means  of  the  capstan. 

That  a  laborer  may  produce  the  best  effect  by  means  of  the  crank- 
handled  windlass,  the  length  of  the  lever  must  not  be  more  than 
from  16  to  18  inches,  corresponding  to  the  length  of  arm  of  the 
laborer,  and  the  axis  of  the  barrel  must  not  be  more  than  36  to  31 
inches  above  the  floor  on  which  the  laborer  stands,  for  men  of  average 
height.  The  handle  of  the  windlass  is  adapted  for  one,  two,  or  more 
men,  according  to  circumstances.  As  a  man  can  work  with  less 


130 


WINDLASS. 


fatigue  while  pushing  and  pressing,  than  while  lifting  and  pulling, 
the  effort  required  at  each  point  of  a  revolution  of  the  handle  is  not 


Fig.  119. 


Fig.  120. 


Fig.  121. 


equal,  and,  therefore,  it  is  well  in  double-handled  windlasses  to  set 
the  handles  180°  apart,  and,  when  more  handles  are  applied,  to  dis- 
tribute them  equally. 

The  day's  work  of  a  man  working  a  windlass  has  been  found  to 
be  1'175040  feet  Ibs.  with  a  mean  effort  K=  17  Ibs.,  and  mean  velo- 
city c  =  2,4  feet,  and  length  of  day  8  hours.  The  calculations  for 
the  windlass  are  the  same  as  for  the  wheel  and  axle. 

If  the  resistance  Q,  Fig.  121,  act  with  the  lever  CB  =  6,  and  the 
power  P  on  the  lever  CA  =  a,  then  Pa  =  Qb ; 
and,  therefore,  the  power  corresponding  to  a 

given  resistance  is  P  =  -  Q.    If,  again,  D  be 
a 

the  pressure  on  the  journals  or  gudgeons  and 
r  the  radius  of  the  gudgeons  CE,  then  Pa  = 

Qb  +fDr,  and  hence  P  =  -  Q  +  -  .fD.    If 
a  a 

the  resistance  Q,  together  with  the  friction  - 

f  D,  consist  of  the  useful  resistance  Q1?  the 
constant  prejudicial  resistance  W,  and  the  variable  prejudicial  resist- 
ance 8  Q,  or,  Q=(l  +  «)  Q,+  W,  then  P  =  b-  [(1+  S)  Q,  +  W}  = 

K-\ —  •  -Q-  and,  therefore,  the  proportion  of  the  winch  and  barrel 

radius  should  be :-  =  - —      '     l      * — .     But    as  the  winch  has  a 
o  K 

prescribed  height  of  16  to  18  inches,  the  leverage  of  the  resistance, 
or  radius  of  the  barrel  is  to  be  determined  by  this,  viz: 

b  = ,  in  order  that  the  laborer  may  work  to  the 

greatest  advantage. 

Example.  On  a  two-handled  windlass  the  resistance  is  200  Ibs.,  viz.:  150  Ibs.  of  use- 
ful resistance,  and  30  Ibs.  constant,  and  20  Ibs.  variable  prejudicial  resistance.  The 
leverage  of  the  resistance  is  4  inches,  that  of  the  power  18  inches,  the  radius  of  the 


VERTICAL  CAPSTAN. 


131 


journal  $  inch,  the  co-efficient  of  friction  /=0,1,  and  the  weight  of  the  barrel  &c  80 
Ibs. ;  required  the  useful  effect  of  such  a  machine.  The  whole  power  required,  if 'the 
pressureonthe  journals  be  taken  200-|-SO=  280  Ibs.,  is  P  =  jL  2004-Ol_J__  280 

IS'  '   2  .  18   ' 

=  44,444-0,5  =  44,94  Ibs.,  and,  therefore,  the  effort  of  each  laborer  must  be  22,47  Ibs 
and,  according  to  Gerstners  formula,  the  velocity  of  the  power,  or  of  the  handle  of  the 
windlass: 

v  =  (2—  -  )  c=  (2 —  )  .  2,4=  1,628  feet,  and  that  of  the  resistance: 

\          Jx  /  \  17    / 

w  =  —  v  =  y  .  1,628  =  0,362  feet,  and  the  useful  effect  per  second : 

b 

Q,M.-  =  0,362  .  150  =  54,3  feet  Ibs.,  and  daily  =  1,563840  Ibs.,  and  the  efficiency  of  such 
an  application  of  the  power  of  two  laborers,  the  day's  work  of  each  of  whom  is  assumed 

to  be  1,175040  : 


Fig.  122. 


Fig.  123. 


2,350080 

§  59.  Vertical  Capstan. — 
When  the  axis  or  barrel  of 
the  windlass  is  vertical,  it  is 
termed  a  capstan,  Fig.  122. 
It  is  chiefly  used  on  land 
for  moving  great  weights  a 
short  distance,  or  for  remov- 
ing great  weights,  as  blocks 
of  stone  from  a  quarry,  or 
for  the  erection  of  obelisks,  &c.  Its  use  on  board-ship  is  well  known. 

•The  horse-capstan,  in  its  different  applications  as  the  prime  mover 
of  mill  work,  or  as  a  whim-gin,  as  it  is  termed,  by  miners,  is  a  modi- 
fication of  the  windlass 
easily  comprehended.  The 
cattle  employed  in  working 
the  vertical  windlass  or  gin, 
go  round  in  a  given  path, 
pushing  or  pulling  at  the 
arm  of  the  machine.  Fig. 
123  shows  the  usual  con- 
struction of  the  whim-gin 
(Fr.  baritel  a  chevaux  ma- 
ndge ;  Ger.  Pferdegb'pel, 
Handgopel).  BO  is  the 
axis,  having  a  pivot  at  0 
resting  in  &  footstep,  ACJl^ 
is  the  double  arm  or  lever, 
with  fork-shaped  shafts  G, 
Gj.  These  cross  the  backs 
of  the  horses,  and  the  harness  is  attached  to  them.  The  resistance 
Q  acts  at  the  circumference  of  a  barrel  or  drum,  or  toothed  wheel 
B,  either  directly  or  indirectly.  The  length  of  lever  is  made  as  great 
as  conveniently  can  be  done,  that  the  animals  may  have  the  largest 
possible  circle  to  move  in.  The  radius  should  not  be  less  than  from 
20  to  30  feet.  The  line  of  traction  must  be  as  nearly  horizontal  as 
possible,  and,  therefore,  the  height  of  the  lever  should  be  fixed, 
according  to  the  height  of  the  animals  working  on  it.  By  the  ar- 


182  VERTICAL  CAPSTAN. 

rangement  shown  in  Fig.  123,  the  horses  or  other  cattle  work  very 
nearly  at  right  angles  to  the  beam  or  lever;  but  if  the  horses  be 
attached  by  traces  to  a  cross  bar  and  hook,  the  direction  of  traction 
makes  a  certain  angle  with  the  beam,  becoming,  in  fact,  a  chord 
of  the  circular  path. 

From  the  length  of  beam  CA  =  a,  Fig.  124,  and  the  length  of 
traces  J1D  =  d,  the  length  of  levers  of  the 
Fig.  124.  horses  is : 


CJV* 


-.-J--4 


or,  approximately,  =  a  —  — — .     It  is  a  re- 
8  ci 

suit  of  experiment,  that  a  man  can  work  eight 
hours  daily  on  the  beam  of  capstan  or  gin, 
exerting  an  effort  of  25J  Ibs.  at  the  rate  of 
1,9  feet  per  second,  and  can,  therefore,  pro- 
duce a  day's  work  =  25.5  .  1,9  .  28800=  T395360  feet  Ibs.;  that, 
on  the  other  hand,  a  horse  working  on  a  gin  for  8  hours  daily,  with 
a  speed  of  2,9  feet  per  second  (a  walk)  can  exert  an  effort  of  95  Ibs., 
or  produce  a  day's  work  =  95  .  2,9  .  28800  =  7'934400  feet  Ibs. 
The  power  is  to  the  resistance,  on  the  capstan  or  gin,  as  for  any 

wheel  and  axle,  or  P  =  Q  -,  when  b  and  a  are  the  arms  or  leverages 
a 

of  the  resistance  Q,  and  power  P  respectively.  The  frictions  at  the 
footstep  and  at  the  periphery  of  the  pivots  at  top  and  bottom  have  to  be 
considered ;  for  these  require  an  increase  of  the  power.  If  G  be  the 
weight  of  the  gin  or  capstan  complete, 
-  *25 and  rl  the  radius  of  the  pivot,  the  statical 
moment  of  the  friction  on  the  footstep  =  f 
/  G  r},  (Vol.  I.  §  171.)  The  point  of  ap- 
plication of  the  resistance  B,  Fig.  125, 
generally  lies  nearer  the  one  pivot,  than 
the  other,  and  thus  the  pressure  on  the 
two  is  different,  and  their  dimensions  are, 
of  course,  proportional  to  the  strain. 

If  the  point  of  application  of  the  resist- 
ance be  at  the  distance  BO  =  lv  from  the 
pivot  0,  and  CB  =  12  from  the  pivot  C,  and  if  the  whole  length  of 
the  upright  shaft  CO  =  I  =  ^  +  /2,  then  the  pressure  on  the  lower 

pivot  Dj  =  j  Q,  and  the  pressure  on  the  upper  pivot  D2  =  -1  Q,  as  is 

manifest  if  we  first  consider  C  and  then  0,  as  the  fulcrums  of  the 
lever  CBO.     Thus,  the  sum  of  the  statical  moments  of  the  lateral 

friction  on  the  pivots  =/Ari  +/  D2r2  =  ^  "t  ^  -/Q>  and  the 
equation  of  equilibrium  for  the  gin,  is 

+^l. 


TREAD-WHEELS. 


133 


Remark  1.  The  application  of  the  whim  gin,  for  drawing  from  mines,  is  treated  of  in 
the  third  section. 

Remark  2.  French  authors  assert  that  a  horse,  going  at  a  trot,  can  work  daily  4£  hours, 
exerting  an  effort  of  30  kilog.  =  66  Ibs.  at  a  speed  of  2  metres  =6,6  feet,  and,  therefore^ 
can  produce  a  day's  work  of  7'055000  feet  Ibs.  If  we  apply  Gerstner's  formula,  and 
put  K=  120  Ibs.,  c  =  4  feet,  v=6,6  feet,  <  =  8  hours,  and  z  =  4£  hours,  we  get  the 

power  P=(% —  )  (2 — ^  .  120=  60  Ibs.,  and,  therefore,  the  day's  work  =  60 

\          4   /  \  8   / 

.  6,6  .  4,5  .  3600  =  6'415200  feet   Ibs.,  or  pretty  nearly  the  result  alluded  to.     If,  how- 
ever, we  take  the  velocity  2,9  feet  of  a  walk  as  the  basis,  we  get  by  Gerstner's  formula 

/         2  9\ 
a  much  greater  effort,  viz  :    (2 —  \  .  120=  153  Ibs.,  and,  therefore,  the  day's  work 

(8  hours)  =  12'778560  Ibs. 

§  60.  Tread-wheels,  or  Tread-mills.  —  The  weight  of  men  and 
cattle  is  sometimes  used  as  the  moving  power  of  machines,  the  effort 
being  exerted  by  climbing  on  the  periphery  of  the  wheel.  The 
wheels  consist  of  two  crowns,  connected  with  an  axle  by  arms,  and 
with  each  other  by  a  flooring.  The  laborer  treads  either  at  the 
internal  or  on  the  external  circumference,  cross  pieces,  as  steps, 
being  provided  for  his  steadier  support  at  intervals  of  1|  feet.  Figs. 

Fig.  126.  Fig.  127. 


Fig.  128. 


126  and  127  represent  the  more  usual  construction  of  tread-mills. 
Fig.  128  is  a  construction  of  wheel 
analogous  to  an  endless  ladder,  but 
is  not  much  used.  On  it,  the  laborer 
is  placed  at  the  level  of  the  axis,  so 
that  his  weight  acts  entirely,  and  with 
the  radius  CA  =  a  surpassing  that 
of  the  wheel  itself.  In  tread-mills 
the  laborer  is  placed  at  an  acute  angle 
ACK  =  a  from  the  summit,  or  the 
bottom  of  the  wheel,  and,  therefore, 
the  leverage  of  his  weight  G,  is  less 
than  the  radius  of  the  wheel  CA  =  a, 
viz:  CJY  =  a,  =  CA  sin.  CAJY=  a 
sin.  a.  But  then  the  fatigue  of  the 

laborer  on  the  endless  ladder  is  greater  than  on  the  tread-mill. 
VOL.  II.— 12 


134  MOVABLE  INCLINED  PLANES. 

the  former  case,  it  is  the  effort  necessary  to  mount  a  vertical  ladder; 
in  the  other,  it  is  that  for  going  up  an  inclination  given  by  the  tan- 
gent AT,  making  the  angle  TAH=  CJ1N  =  a.  The  effort  P  in  the 
case  of  the  ladder  is,  therefore,  G,  while  in  the  tread-mill  it  is  G 
sin.  a.  If  the  resistance  Q  act  with  the  leverage  CB  =  6,  then  for 
the  ladder-wheel  Ga  =  Qb,  while  for  the  tread-mill  Ga  sin.  a.  =  Qb, 
by  substituting  the  power  or  effort,  as  in  the  wheel  and  axle  ;  Pa=  Qb. 
Mathematically  considered,  therefore,  the  tread-mill  gives  no  ad- 
vantage over  the  windlass  or  capstan  ;  but  the  laborer  can  produce 
a  much  greater  day's  work  by  the  one  than  by  the  other,  and,  there- 
fore, they  are  often  advantageously  employed.  The  application  of 
four-footed  animals  on  these  wheels  is  inconvenient,  and  not  advan- 
tageous in  any  point  of  view. 

It  has  been  deduced  from  experiment  that  a  man  can  work  near 
the  centre  of  the  wheel,  i.  e.,  near  the  level  of  the  axis  for  8  hours 
daily,  exerting  an  effort  of  128  Ibs.,  and  going  at  0,48  feet  per 
second,  while  he  can  work  for  the  same  time,  exerting  an  effort  of 
2of  Ibs.,  and  going  at  2£  feet  per  second,  when  his  position  is  24° 
from  the  vertical.  In  the  one  case,  the  day's  work  amounts  to 
1769000  feet  Ibs.,  and  in  the  other  1'663000  feet  Ibs.  Horses  and 
other,  cattle  produce  less  effect  on  such  machines  than  by  means  of 
a  gin.  A  part  of  the  advantage  arising  from  the  use  of  tread-wheels 
is  lost  in  the  increased  friction  of  their  axles  beyond  that  of  wind- 
lasses or  capstan.  If  n  G  be  the  weight  of  the  laborers,  Gl  the 
weight  of  the  machine,  and  if  the  resistance  Q  act  vertically  down- 
wards, the  pressure  on  the  journals  D  =•  n  G  +  Gj  +  Q,  and  if  r  be 
the  radius  of  the  journal,  the  moment  of  friction  =f  (n  G+  Gl+  Q) 
r,  and  the  ratio  of  power  to  resistance  is:  n  G  a  sin.  o  =  Qb  -f-  f 
(n  G  +  G,  +  Q)  r. 

If  the  resistance  be  given,  the  angle  of  ascent  may  be  determined, 
viz.: 

sin  a  =  Q6+/(nG+G1+Q)r 

n  Ga 
or  the  number  of  laborers 

„_  <?&+/(£:+  Q)r 
G(asin.  o  —  fr) 
Men  work  to  the  greatest  advantage  when  their  effort:  nP=n  G 

Fig.  129.  sin,  a==  n  K  -f  -  .  _  ,  or  when  sin.  *  = 


§  61.  Movable  Inclined  Planes.  —  For 
farming  purposes,  in  breweries,  &c.,  the 
arrangement  sketched  in  Fig.  129,  is 
sometimes  applied.  The  horse  or  ox 
works  on  such  an  inclined  plane  for  short 
spells.  The  machine  has  this  advantage, 
that  the  animal  may  be  left  without  a 


WATER-COXDUITS.  135 

driver.  The  action  of  the  animals  is  in  every  respect  the  same  as 
in  tread-mills,  when  they  work  near  the  horizontal  radius.  The 
machine  consists  of  a  shaft  BO,  the  axis  of  which  is  inclined  20°  to 
25°  from  the  vertical,  and  of  a  plane,  from  20  to  25  feet  in  radius, 
set  at  right  angles  to  the  shaft,  and,  therefore,  having  an  inclination 
of  20°  to  25°  to  the  horizon.  If  the  animal  moving  the  machine 
work  at  a  distance  CA  =  a  from  the  axis  of  the  shaft,  and  if  the 
angle  of  inclination  of  the  plane,  or  the  inclination  upon  which  the 
animal  may  he  supposed  to  be  moving  =  0,  then  the  power  P  =  G 
sin.  a,  and,  therefore,  the  moment  of  rotation  =  Pa  =  Ga  sin.  a. 

If  the  resistance  be  applied  with  a  leverage  6,  its  moment  is  Q  6; 
and  if  G1  be  the  weight  of  the  machine  when  in  work,  and  r  be  the 
radius  of  the  pivot,  the  moment  of  friction  on  the  footstep  =  f/(G 
+  G,)  cos.  a  .  r,  and  the  moment  of  friction  on  the  periphery  of  the 
pivots  =f  (G-f  Gl —  Q)  sin.  a  .  r;  because  the  weight  G  +  G1  re- 
solves itself  into  the  components  (G  +  Gj)  cos.  a,  in  the  direction  of 
the  axis,  and  (G  +  Gt)  sin.  a,  in  the  direction  of  the  inclination  of 
the  plane,  whilst  Q  acts  in  the  opposite  direction  to  this  latter. 
Whence  follows 
G  a  sin.  «*=  Q  (b— fr  sin.  a)  +f(G  +  GJ  (§  cos.  «  +  sin.  a)  .  r. 

Example.  How  many  men  are  required  to  be  put  upon  a  tread-mill  of  20  feet  diame- 
ter, in  order  to  raise  a  weight  of  900  Ibs.,  acting  with  a  leverage  of  0,8  feet?  If  we 
estimate  the  weight  of  the  wheel,  and  its  load  at  5000  Ibs.,  and  taking  the  radius  of  the 
pivot  at  2^  inches,  and  the  co-efficient  of  friction  at  0,075,  then  the  statical  moment  of 
(Tie  resistance  =0,8  .  900  +  0,075 .  /s  .  5000  =  720  +  78  =  798  feet  Ibs.,  and,  therefore, 

the  power  at  the  circumference  of  the  wheel  =  —  =  79,8  Ibs.     A  laborer  placed  24° 
back  from  the  summit  of  the  wheel,  exerts  an  effort  of  25$  Ibs.,  and,  therefore,  the  num- 

VQ  Q 

ber  of  men  required  is  _!_=  3.      These  men  could  produce  3  .  1G63000  =  4989000 
feet  Ibs.  per  day  of  8  hours,  and,  therefore,  they  could  raise  the  weight  Q  daily  through 

/1QKQOOO 

/*       u       =  5890  feet  high  ;  or,  supposing  the  load  had  to  be  raised  only  200  feet  high, 
900 

the  three  men  could  raise  £^2  =  30  times  900  Ibs.  to  the  height  of  200  feet. 
200 


CHAPTER    III. 

ON  COLLECTING  AND  LEADING  WATER  THAT  IS  TO  SERVE  AS  POWER. 

§  62.  Water-conduits. — Water  that  is  to  serve  as  power  (Fr.  I'eau 
motrice;  Ger.  Aufschlagewasser),  to  be  applied  to  machines,  is  col- 
lected from  streams  and  rivers,  or  from  springs.  In  most  cases  the 
machines  have  to  be  erected  at  some  distance  from  the  point  at  which 
the  water  can  be  collected,  and  must  be  led  to  the  machine  in  what 
is  termed  the  lead  or  lete,  or  water-conduit,  or  water-course. 

The  lead  may  either  be  an  open  channel  or  canal  (Fr.  canalet, 


136  SWELL,  OR  BACK-WATER. 

rigoles],  or  it  may  be  a  close  pipe  (Fr.  tuyaux  de  conduite;  Ger. 
Rohrenleitungen}.  Pipes  are  best  adapted  for  smaller  quantities  of 
water.  They  have  this  great  advantage,  that  they  may  be  led  in 
any  way  within  the  hydraulic  range  of  variation  of  level,  whilst 
canals  as  letes,  must  have  a  continuous  fall.  Valleys  and  hills  may 
often  be  passed  by  pipes  without  trouble  or  expense,  while  the  open 
channel  requires  the  cutting  of  drifts  or  tunnels,  and  the  erection  of 
aqueducts. 

§  63.  Dams. — The  vis  viva  of  running  waters,  of  brooks  and  rivers 
having  velocities  of  from  1  to  6  feet  per  second,  is  seldom  sufficient 
to  allow  of  their  direct  application  as  power  to  drive  machines.  To 
increase  the  vis  viva,  or  to  bring  the  weight  of  the  water  into  action, 
it  is  necessary  to  dam  it  up  to  create  a  head  or/all  (Fr.  chute;  Ger. 
Grefalle).  Water  is  dammed  up  by  weirs,  dams,  or  bars  (Fr.  barrages  ; 
Ger.  Wehre). 

Weirs  are  either  overfall  weirs,  or  they  are  sluice  weirs.  Whilst 
in  the  former  the  water  flows  freely  over  the  saddle-beam  cill,  or 
highest  edge  of  the  weir,  in  the  latter  movable  sluice-boards  dam  the 
water  above  the  summit  of  a  weir,  which  may  be  either  natural  or 
artificial.  The  overfall  weir  is  usually  laid  down  with  the  view  of 
constraining  a  portion  at  least  of  the  water  of  a  river  or  stream  to 
enter  a  side  canal  above  it,  or  a  lete  by  which  it  is  conducted  to  the 
machine  by  which  the  power  of  the  water  is  to  be  applied ;  and  the 
sluice  weir,  is  used  when  the  object  is  to  get  an  increased  vis  viva  to 
the  water,  which  is  then  directly  applied  to  a  machine  immediately 
below  the  weir. 

In  large  rivers,  dams  are  frequently  built  to  occupy  only  a  part 
of  the  width  of  the  stream.  These  dams  are  termed  incomplete  wiers, 
in  contradistinction  to  complete  weirs,  which  are  laid  from  side  to 
side  of  the  stream.  The  piers  of  bridges  are  examples  of  incomplete 
weirs  (Fr.  barrage  discontinus ;  Ger.  Lichte  Wehre},  contracting 
the  passage  for  the  stream  to  a  certain  extent. 

Overfall  weirs,  too,  may  either  be  complete  or  imperfect.  The 
summit  of  the  complete  overfall  rises  above  the  surface  of  the  water 
in  the  part  of  the  stream  below  it,  whilst  the  top  of  the  incomplete 
weir  lies  below  that  level,  so  that  a  part  of  the  water  flowing  over 
undergoes  a  resistance  from  the  water  below-weir. 

§  64.  Swell,  or  Back-water. — Any  of  the  constructions  we  have 
above  alluded  to,  dam  back  the  water,  produce  a  swell  above  the 
weir,  an  elevation  of  the  water's  surface,  and,  therefore,  a  decrease 
of  velocity.  The  height  and  amplitude  or  extent  backwards  to  which 
this  rise  of  the  water  surface  extends,  is  a  matter  important  to  be 
determined  with  reference  to  the  dimensions  of  the  weir. 

A  knowledge  of  this  relation  between  the  weir,  and  its  eifects  on 
the  river  above  it,  is  not  only  necessary  because  by  damming  up  the 
water  too  high,  we  should  involve  the  district  above  in  floods  to  which 
they  had  not  been  previously  subjected,  but  we  may  interfere  with 
other  establishments,  robbing  them  of  a  part  of  their  fall,  by  throw- 
ing back-water  upon  them.  The  level  of  the  summit  of  weirs  is  often 


CONSTRUCTION  OF  WEIRS.  137 

fixed  by  law  or  prescription  according  to  a  standard  peg,  or  mark  ; 
any  alteration  of  which  is  an  offence  liable  to  penalties  (see,  in  refe- 
rence to  the  English  law  o%n  this  subject,  "Fonblanque  on  Equity"). 
The  peg  generally  has  a  scale  attached  to  it,  by  which  the  supply  of 
water  may  be  read  off  at  a  glance. 

The  water  flowing  over  an  overfall,  or  through  an  incomplete 
weir,  acquires  a  waving  eddying  motion,  the  action  of  which  is  very 
severe  on  the  bed  of  the  river  immediately  below  the  weir,  so  that 
particular  arrangements  have  to  be  made  in  the  erection  of  weirs  to 
withstand  this  action. 

The  quantity  of  water  contained  in  or  flowing  through  streams  or 
rivers,  is  different  at  different  times,  so  that  we  have  the  expressions 
full,  average,  and  dry,  applied  to  the  state  of  rivers,  corresponding 
in  Britain,  to  winter,  autumn,  and  summer,  though  not  very  definitely 
fixed  as  to  the  particular  period  of  the  seasons.  It  is  evidently  ne- 
cessary to  have  accurate  information  as  to  the  mean  supply  of  water 
yielded  by  a  brook  or  stream,  proposed  to  be  applied  as  water  power. 
The  state  of  the  stream  in  autumn  and  spring  may  be  taken  as  the 
mean  state,  but  for  any  important  undertaking  of  this  nature  a  series 
of  hydrometrical  observations  should  be  instituted,  that  the  question 
of  the  supply  of  water  may  be  accurately  determined.  Any  one  of 
the  methods  discussed  in  Vol.  I.  §  376,  &c.,  may  be  adopted  for  this 
purpose. 

,  §  65.  Construction  of  Weirs. — For  obtaining  water  power,  the 
overfall  weir  is  the  most  important  means.  Weirs  are  built  either 
square  across  the  stream,  or  inclined  to  the  axis  of  it.  They  are 
often  built  in  two  parts  inclined  to  each  other,  the  angle,  which  is 
laid  up-stream,  being  rounded  or  not ;  they  are  formed  as  polygons 
also,  and  as  segments  of  .a  circle,  the  convexity  being  always  turned 
to  the  stream.  Weirs  are  built  of  wood,  or  of  stone,  or  of  both  com- 
bined. They  have  frequently  to  be  founded 
on  piles,  from  the  difficulty  of  getting  a  Fie-  13°- 

sound  foundation.  The  cross  section  of 
wooden,  or  other  dams,  is  more  or  less  of 
the  form  of  a  five-sided  figure  J1BCDE, 
Fig  130,  in  which  JIB  is  termed  the  breast, 
BC  the  front  slope,  CD  the  apron,  DE  the 
back,  and  EA  the  sole,  and  C  the  saddle  or 

cill.     The  cross  section  of  stone  weirs  is  generally  composed  of 
curved  lines,  as  regards  the  apron  and  back,  the  object  being  to 
get  the  rush  of  water  smooth- 
ly away  from  the  foot  of  the 
apron,  so  as  to  prevent  corro- 
sion in  time  of  floods. 

An  overfall  weir,  such  as 
is  represented  in  Fig.  131, 
consists  of  a  row  of  piles  D, 
going  across  the  stream,  and 


13* 


CONSTRUCTION  OF  WEIRS. 


a  walling-piece,  or  saddle-beam  C  on  the  top  —  of  walling  E  in  front 
of  the  piles  —  a  second  row  of  piles  F  further  down-stream  and 
parallel  to  the  first  —  of  a  casing  of  hard  laid  pavement  G,  between 

the  two,  and  which  is  con- 
132.  tinued  onwards  with   the 

same  curvature,  forming  an 
apron  (which  should  be 
continued  so  that  it  turns 
slightly  upwards).  The 
weir  in  Fig.  132,  shows 
the  manner  of  founding  on 
piles,  the  intervals  between 
the  piles  being  cleared  out 
as  far  as  possible,  and  rammed  with  concrete,  and  upon  this  the 
superstructure  is  raised. 

The  construction  of  wooden  wiers  is  sketched  in  Fig.  133.     AE 


Fig;.  133. 


is  a  wall  of  beams,  lying  tight,  one  on  the  other,  on  the  top  of 
which  comes  the  saddle-beam  A.  These  beams  are  confined  by  a 
double  row  of  piles  CD  and  CrDv  and  the  piles  EF  and  G//,  driven 
as  breast  and  back  of  the  dam,  form  resting  points  for  the  planking 
of  the  dam.  The  interior  of  the  dam  is  filled  with  stone,  clay,  con- 
crete, or  such  material.  The  apron  K{  of  the  dam  is  continued 
onwards  as  substantially  as  possible,  in  the  manner  shown  in  the 

sketch.     This  latter  is  a 

134-  point  of  great  importance. 

At  L  the  sluice  of  the  lete 
is  visible.  A  submerged 
weir  is  shown  in  Fig.  134. 
A  is  the  saddle-beam,  JIB 
are  the  guide-columns,  in 
grooves,  in  which  the 
sluice  works.  The  ar- 
rangements for  raising  or 
opening  and  lowering,  or 
shutting  the  sluice  are 
various.  A  capstan-like  arrangement  is  shown  in  the  figure,  the 


HEIGHT  OF  SWELL. 


139 


sluice-board  hanging  by  chains.  The  piles  in  such  a  construction 
must  be  cleared  for  some  depth,  and  the  interstices  well  rammed 
with  puddle  or  concrete,  to  prevent  leakage. 

§  66.  Height  of  Swell.  —  By  aid  of  the  hydraulic  formulas  we  have 
investigated  (Vol.  I.),  the 

height  and  amplitude  of  F'R-  135. 

the  back-water  for  any 
given  dam  may  be  easily 
determined.  If,  in  the 
case  of  a  dam  represented 
in  Fig.  135,  h  be  the  head 
JIB,  and  if  b,  be  the 
breadth,  and  k,  the  height 
due  to  the  velocity  c  of 
the  water  as  it  flows  up  to  the  weir,  or: 

A;  =  —  ,  theu  the  quantity  of  water  discharged  by  the  weir  is  (Vol. 

I.  §  321)  Q  =  f  p  b  v/2Jr[A  +  kfi  —  A'].  If,  on  the  other  hand, 
the  quantity  discharged  be  known,  the  head  corresponding  to  it  upon 

the  saddle-beam:  A  =  (  —  *         -f  k*  y  —  &•      In  order,  therefore, 

*         ' 


height  B0  =  x  of  a  weir  to  produce  a  given  head,  or  rise 
of  the  water  surface  at  the  weir  =  A,,  we  put  AC  +  CO  =  JIB  +  BO, 
or,  if  the  original  depth  of  the  water  down-stream  CO  be  put  =  a, 
then  A,  -f  a  =  h  4-  #,  and  hence  x  =  a  +  At  —  h. 

When  the  back-water  or  head  raised  is  considerable,  say  x  =  at 
least  2  feet;  the  velocity  of  the  water,  as  it  comes  to  the  weir  &,  may 
be  neglected,  and,  therefore,  we  may  put: 


and  according  to  experiments  of  the  author,  the  co-efficient  /*  may  be 
taken  =  0,80  for  this  case. 

In  the  case  of  the  submerged  weir,  Fig.  136,  the  calculation  is 
somewhat  more  complicated, 
because  in  this  case  two  dif- 
ferent discharges  are  com- 
bined. The  height  JIC  =  h 
of  the  water  above  the  saddle- 


beam  is  greater  in  this  case 

than  the  height  JIB  =  hv  to 

which  the  water  is  raised  by 

the  darn,  and,  therefore,  only 

the  water  above  the  level  B  flows  away  freely,  whilst  the  water  ande 

B  flows  away  under  the  head  or  pressure  JtB  =  h,.     The  d 

throuh 


and  that  through  BC  -*  h  —  A,,  is: 


140  HEIGHT  OF  SWELL. 


and  consequently  the  whole  quantity,  or, 

Q,  +  Q2  =  Q=  ,6^%[f  [(A,  +  *)S  — 

From  the  quantity  of  water  Q,  and  the  height  At  to  which  the  water 

is  raised,  we  have  the  height  of  water  above  the  saddle  : 


(*,  +  *) 

from  which  the  height  of  weir  CO  =  x=a-}-hl  —  h  may  be  deduced. 
It  is  evident  that  h  >  A,  or  the  weir,  is  a  submerged  or  imperfect 
weir,  when 

Q>  t^v^rft**)1—  fc1]- 

Example.  A  stream  of  30  feet  width,  and  3  feet  in  depth,  discharges  310  cubic  feet 
of  water  per  second.  It  is  required  to  raise  it  4^  feet  by  means  of  a  weir.  What  height 
of  weir  is  necessary?  As  the  height  of  the  water  to  be  raised  is  considerable  in  this 
case,  we  may  confidently  use  the  simpler  formula: 

x  =  a+A,—  (  _  3Q  _)*.     In  this   formula  a  =  3,  A,  =  4,5,  Q=310,  6  =  30, 
V2  / 


H  =0,80,  and  ^/Qg  =  8,02  for  the  case  in  question.     Hence  : 

x  =  3  +  4.5  —  (  3  •  10  _  ^  =  5,7  feet:  and.  therefore,  the  overfall  is  a  perfect 

\2  .  0,8  .  30  .  8,02  / 

weir,  as  was  presumed.  If  it  were  required  to  raise  the  water  up  only  2  feet,  x  would 
be  3,2  feet,  or  the  weir  would  still  be  perfect.  If  1^  feet  only  were  required,  the  dam 
would  not  require  to  rise  above  the  level  of  the  water  down-stream,  or  the  natural  level 
of  the  water  in  the  stream  ;  and  would  be  a  submerged  weir.  Applying  the  complete 
formulas  to  this  case,  and  putting 


\4,5.30/ 
0,0155.  5,27=0,084  feet,  and  taking^  again  =  0,80  we  get: 

A  —  *t=  31°  _j    (1,884)*-  (0,084)* 

0,8.30.8,02^/1,584  ]j584$ 

=  1,28  —  1,06  +  0,01  =0,23  feet. 

The  saddle  overfall  must,  therefore,  be  about  i  foot,  or  3  inches  under  the  surface  of 
the  water  on  the  lower  side  of  the  weir,  and,  therefore,  the  height  of  the  weir  itself 
x  =  a+A  —  A,  =  3,25  feet. 

§  67.  The  height  and  amplitude  of  the  back-water  in  the  case 
of  sluice  weirs  may  be  determined  according  to  the  theory  of  the 
discharge  by  sluices.  Three  cases  may  occur.  Either  the  water 
flows  away  unimpeded,  or  it  flows  under  a  counter  pressure  of 
water,  or  it  flows  partly  unimpeded,  partly  under  water.  In  the 
case  of  a  free  discharge,  as  in  Fig.  134,  the  velocity  of  discharge 
depends  upon  h  above,  measured  from  the  centre  of  the  opening  to 
the  water's  surface.  If,  then,  a  be  the  height  of  opening,  and  b 
the  breadth,  then  Q  =  ^  a  b  ^/'2  gh,  and,  therefore,  inversely 

A  =  2r~  (  —  ii  '  or>  taking  into  consideration  the  velocity  k  with 
2g  \pab/ 

which  the  water  comes  up  to  the  sluice, 

A  =  —  (-^T  )  —  %•   For  the  height  of  opening,  we  have  the  formula  : 


HEIGHT  OF  SWELL. 


,  or,  if  7^  =  the  height  to  which  the  water  is  raised  by 

the  dam  above  the  sill,  be  given : 
n 

•     According  to  the  author's  experiments  p.  is 


here  =  .60. 

If  the  under- water  lie  back  to  the  sluice,  as  in  Fig.  137,  then  the 
difference  of  level  J1B  =  h,  is  the  head  to  be  introduced  as  pressure 
in  the  above  formula.  In  this  case,  therefore,  the  opening  corre- 
sponding to  a  given  head  h  is  :  a  =  " 

M  b  ^/2gh 

When  the  level  of  the  under-water  is  within  the  range  of  the 

Fig.  137.  Fig.  138. 


sluice's  opening,  as  shown  in  Fig.  138,  one  part  flows  away  unim- 
peded, whilst  the  other  flows  under  water.  If  the  height  of  the 
water  is  raised,  or  the  difference  of  level  AB,  Fig.  138=  Arthe  height 
BC  of  the  part  of  orifice  of  discharge  above  the  surface  of  the  water 
=  «j,  and  BD  the  height  of  the  part  under  this  surface  =  a2,  then 
the  quantity  of  water  for  the  former  part:  3?C_ 


Q1=fta1b    hg(h  —  a-±-\  and  for  the  other: 

Q2  —n,  a2  b  \/2gh,  therefore,  the  whole  quantity  : 


From  the  quantity  of  water  discharged  Q,  the  height  to  which  the 
water  is  raised  A,  and  the  depth  a2  of  the  sill  or  saddle  of  the  weir 
under  the  under-water  surface,  we  deduce  the  distance  of  the  sluice- 
board  from  this  surface: 


Example  1.  How  high  must  the  boards  of  the  sluice  weir,  Fig.  134,  be  raised,  which 
has  to  let  off  200  cubic  feet  of  water  per  second,  the  breadth  b  being  =  24  feet,  and  the 
height  h,,  to  which  the  water  is  dammed  above  the  sill  =  5  feet?  In  the  case  of  unim- 
peded discharge: 

250  2,16 


142 


DISCONTINUOUS  WEIRS. 


approximately : 
ing  required:  a 


1,  hence :     /5  _  °_  __  ^4,5=  2,12,  therefore,  the  height  of  open- 
>/          2 

* 
1,02  feet  =  J2.24  inches. 


2,16 
2,12 


Example  2.  What  amount  must  the  sluice,  Fig.  137,  be  drawn  up,  in  order  that  it  may 
discharge  120  cubic  feet  of  water  per  second,  under  a  head  of  J,5  feet,  the  width  of  open- 
ing being  30  feet?     This  is  a  case  of  discharge  under  water,  therefore, 
120 

a  =  --  —=  0,678  feet  =  8,14  inches. 


0,6  .  30  . 

Example  3.  It  is  required  to  determine  the  quantity  of  water  which  flows  through  a 
sluice  opening  (Fig.  138)  of  breadth  b  =  18  feet,  height  CD  =  a,  +  o,  =  1,2  feet, 
•when  the  head  jlB  —  2  feet  =  h,  and  the  height  of  water  above  the  sill,  aa  =  6,5  feet. 
In  this  case  p  b  ^/Tg  =  0,6  .  18  .  8,02  s=  86,6.  Further  a,  ^/h  =  0,5  </2  =  0,707, 

and  a,     \h  —  —  '  =  0,7  y/1,65  =  0,899,  therefore,  the  quantity  of  water  required  Q  = 
86,6  (0,707  +  0,899)  =  86,6  .  1,606  =  139,07  cubic  feet. 

§  68.  Discontinuous  Weirs.  —  The  height  of  the  back-water  in  the 
case  of  incomplete  or  discontinuous  weirs,  such  as  piers  of  bridges, 
jetties,  &c.,  may  be  calculated  in  very  much  the  same  way  as  that 
for  overfalls.  For  the  jetty  BE,  Fig.  139,  there  results  a  damming 
back  of  the  waters,  because  the  stream  is  contracted  from  the  width 
JiC  to  JIB.  If,  therefore,  the  lead  be  closed,  which  it  is  well  to 


Fig.  139. 


Fig.  140. 


assume,  the  whole  of  the  water  of  the  stream  Q  must  pass  through 
the  contracted  passage  JIB.  If  we  put  the  width  JIB  —  6,  the  height 
of  dammed  water  =  J1B1  =  h,  Fig.  140,  and  the  depth  BlCl  of  the 
under-water  =  a,  then  the  quantity  flowing  freely  above  the  under- 
water is  Qj  =  §  n  b  1/2  gh3,  and  the  quantity  flowing  away  as  under- 
water =  Q2=j*ba  <,/'2gh.  Therefore,  the  whole  quantity  going 
away  :  Q  =  p.  b  */2  g  h  (f  h  +  a).  Hence,  inversely,  the  breadth 
of  weir  corresponding  to  given  height  h  of  dammed  water,  is 

--     If  the  height  of  back-water  h,  be  small, 


b  = 


or  the  velocity  of  the  water  great,  the  velocity  of  the  water  as  it 
comes  up  to  the  jetty,  must  be  taken  into  consideration.  If  k  be 
again  taken  to  represent  the  height  due  to  the  velocity  of  the  water 
as  it  comes  to  the  weir,  we  have  : 

ky  —  &*],  and  Q2 


Q,  =  §  n 
and,  therefore  : 


DISCONTINUOUS  WEIRS. 


143 


and  inversely  : 


fy  [f  [( 


[(A  +  k}*  —  *]  +  a  (h  +  ) 

Whilst  in  the  unimpeded  motion  of  water  in  river  channels,  the 
velocity  is  greatest  at  the  surface,  and  decreases  gradually  as  we  go 
downwards  in  the  vertical  depth,  the  case  is  different  when  the  water 
is  dammed  up  by  any  obstruction  in  the  stream.  Then  the  velocity 
increases  from  the  surface  of  the  upper-water  down  to  that  of  the 
under-water,  and  diminishes  very  little  from  thence  downwards  to 
the  bottom.  There  is,  therefore,  a  change  of  velocity  as  represented 
by  the  arrows  in  Fig.  140.  This  must  necessarily  be  the  case,  be- 
cause the  water  above  the  under-water  surface  flows  away  under  a 
pressure  or  head  increasing  from  0  to  ^,  and  the  water  under  it, 
flows  away  under  the  constant  pressure  A,  whilst  for  unimpeded 
motion,  the  pressure  or  head  at  all  depths  =  0.  This  formula  is 
likewise  applicable  in  the  case  of  bridge  piers,  if  b  be  put  to  repre- 
sent the  sum  of  the  openings  between  the  piers.  In  order  to  prevent 
as  much  as  possible  injurious  effects  from  the  eddying  motion  of  the 
water  behind  and  in  front  of  the  piers,  the  starlings  are  added,  pre- 
senting a  rounded  or  angled  prow  to  the  water.  If  the  starling  of 
ti^e  piers  be  round,  or  form  a  very  obtuse  angle,  then  p  is  to  be  taken 
=  .90,  if  the  angle  be  acute  /*  =  .95,  and  if  the  acute  angle  be 
formed  by  the  meeting  of  two  elliptical  or  circular  arcs,  as  in  Fig. 
141,  ft.  becomes  even  .97,  or  very  nearly  1. 


Fig.  141. 


Fig.  142. 


Remark.  If  a  jetty,  or  other  building  contracting  a  stream,  does  not  reach  above  the 
surface,  the  whole  quantity  of  water  Q  may  be  considered  as  composed  of  3  parts.  It 
the  top  of  the  construction  be  beneath  the  under-water  surface  CD,  Fig.  142,  then  the 
quantity  of  water  flowing  away  through  the  section  J1BDC,  is: 


h  being  the  height  of  the  back-water,  and  b  the  breadth  AB. 

Secondly,  the  remaining  part  above  the  top  of  the  building,  and  under  the  constant 
head  A,  or  Qa  =  p  b,  (a  —  a,)  v^g(A+A),  where  a=GH  the  depth  of  under-water, 
6,  =  the  breadth  EF  of  the  building,  and  a,  =  its  height  EH. 

Lastly,  the  part  flowing  away  at  the  end  of  the  building  under  the  constant  n< 
Qz  =  fjtb2a  v/vig  (h  -f-  /fc),  b2  being  the  free  width  CD.     Thus: 

Q  =  $t*b^/Tg  f(h  +  jfcjl_jfcf|  +  M  [ba  —  b,  a.V^gC/i-M),  and,  therefore,  we  can 
calculate  the  length  and  height  of  building  necessary  to  produce  a  given  amoiin 
If,  on  the  other  hand,  ClDl  be  the  under-water  surface,  or  if  the  constructs 
the  surface, 


144 


AMPLITUDE  OF  THE  BACK-WATEK. 


Q 


[(a  -f  h  —  a,  +  *)i  — 


(A  +  A). 

Example.  What  width  BC  must  be  given  to  the  dam  BE,  Fig.  139,  in  order  that  the 
river,  which  is  550  feet  wide,  and  8  feet  deep,  and  delivers  14000  cubic  feet  of  water 
per  second,  may  be  dammed  up  0.75  feet  ? 


and  if  (* 


0,0155  ("14000V  =  0,0155  .  3,18'=0,156, 

\5ftO.  87 
,  then  the  width  of  the  contracted  stream: 


14000 


09  .  8,02  [$  (0,906?  —  156*)  +  8  .  0,906^], 
54000  14000 


238,5  feet, 


7,2 1 8  (0,522  +  7,608,)          7,2 1 8  .  8, 1 3 
and,  therefore,  the  length  or  projection  of  the  dam  =  550  —  238,5=311,5  feet. 

§  69.  Amplitude  of  the  Sack-water. — We  have  now  to  resolve  the 
other  important  question.  According  to  what  law  does  the  height 
of  the  dammed  water  diminish  in  stretching  back,  up  stream  ?  With- 
out having  resort  to  any  peculiar  theory,  this  problem  can  be  solved 
by  the  theory  of  the  variable  motion  of  water  in  river  channels,  ex- 
plained Vol.  I.  §  369,  §  370. 

Let  us  suppose  the  length  of  river  on  which  back-water  from  the 
dam  J1BK,  Fig.  143,  is  perceptible,  divided  into  separate  lengths, 

Fig.  143. 


and  let  us  submit  each  length  separately  to  calculation.  If  a0  be  the 
depth  of  water  J1R  at  the  weir,  a1  the  depth  DE  at  the  upper  end  of 
such  a  length;  J1BDE,  F0the  section  of  the  flowing  water  at  the  weir, 
F1  the  section  at  DE,  Q  the  quantity  of  water,  p  the  mean  circum- 
ference of  the  section  for  this  length,  and  a  the  angle  of  inclination 
of  the  river's  bed,  then,  from  (Vol.  I.  §  370)  the  length  of  the  first 
division,  (a0  and  av  and  F0  and  Fl  being  substituted  for  each  other)  is: 


,_ 


sin.  a  — 


If  a2  be  the  depth  of  water  GH  at  the  upper  end  of  a  second  length 
DEGH,  F2  its  section,  and  pl  the  mean  perimeter  of  the  water  sec- 
tion of  this  part,  then  its  length 


DH=L  = 


AMPLITUDE  OF  THE  BACK-WATER.  145 

Continuing  in  this  manner,  namely,  assuming  arbitrary  decreases  of 
depth  a0 — av  ox — o2,  o2 — a3,  &c.,  and  calculating  from  this  the  sec- 
tions Fv  F2,  F3,  &c.,  and  the  mean  perimeters,  we  get  by  the  for- 
mula, the  distances  /,  lv  J2,  corresponding,  or  the  distances  /,  /  +  / , 
/  +  /t  +  Z2,  &c.,  from  the  weir. 

To  find  the  depth  y  corresponding  to  a  given  distance  x,  we  may 
either  apply  the  method  of  interpolation  to  the  values  l,l  +  lv  l-\-  lt 
+  Z2,  &c.,  just  found,  or  we  may  make  use  of  this  other  formula) 
likewise  given,  Vol.  I.  §  370,  viz. : 


1  — -.  12- 

If  we  put  in  this  instead  of  60,  the  breadth,  and  instead  of  p0,  the 
perimeter,  and  for  v0  the  velocity  at  the  weir,  this  formula  gives  the 
decrease  (a0 — at)  of  the  height  of  back-water  on  the  first  short  length 
/,  and  for  a  next  following  short  length  Jt  this  decrease  is : 


d  — {T 

and,  lastly,  for  a  given  distance  I  4- ^  +  ',4-  •  •  -the  depth:  o0  — 
(«0  —  Oj) — (al — as)  — .  .  .  may  be  calculated. 

Example  1.  A  weir  is  to  be  built  in  a  river  80  feet  wide,  4  feet  deep,  and  discharging 
1400  cubic  feet  per  second,  in  order  to  dam  up  the  water  3  feet  high.  Required  the 
relative  amount  of  damming,  at  distances  back  from  the  weir.  Without  the  dam,  the 

velocity  of  the  water  c  =  =  4,375  feet,  and,  therefore,  according  to  the  table,  Vol. 

8.04 
I.  p.  447,  the  co-efficient  of  resistance  £=0,00747,  and  the  inclination  of  the  channel 

sin.  a.  =  0,00747  .  /!  .  -f!.     If,  therefore,  p  =  84,  .F  =  80  .  4  =  320,  c  =  4,375,  and  J_ 

=  0,0155,  then  the  inclination: 

sin.  a  =  0,00747  21  .  0,0155  (4,375)'  =  .0,0005818,  or,  .00058, 

near  enough.  The  depth  of  water  close  to  the  weir  is  4  +  3  =  7  feet,  and  we  shall 
now  determine  the  distances  at  which  the  depths  64,  6,  5$,  and  5  feet  occur.  If  we 
first  introduce  into  the  formula: 

1—  \*V       F£'  *S         .  fl)1  —  a,  =  0.5. 1^  =  80.7=560, 


=  80  .  6,5  =  520, °Q  =  1400, °«n.  a  =  .00058,  p  =  86,  and  then  for  the  mean  velocity 

2Q     =  ^£2  —  2,59  feet,  £  =.0075,  the  value  of 
F+/\       1080 
0,5—  (0,0000036982—0,0000031888)  .  30434 

0,00058  —  0,0075-^-  (0,0000036982+0,0000031888)  .  30434 
0.5  —  0,0155 


0,00058  —  O.OOU128       0,000452 

To  find  the  distance  back  at  which  a  depression  of  1  foot  in  the  water  a  surlaci 
we  must  again  put  «0  _  a,  =  0,5,  but  F0  =  520,  Ft  =  80  .  6  =  480,  p  =  85,5  and  the 
VOL.  II.— 13 


14(5  BACK-WATER  SWELL. 

mean  velocity  280°  =  2,80  gives  £=0,00749.     Hence,  by  means  of  the  same  formula 
as  above,  we  ge^or  the  distance  in  which  the  surface  lowers,  so  that  the  depth  becomes 


6  feet  i--adOf6,5,  '  ^  0^845  ^  1142  feet. 


.00058  —  .00749 122.  .0,0000080385. 30434 
The  water  at  a  distanced  071  +  1142  =  2213  fee^s,  therefore,  only  6   feet^ deep  or 
.     ,    -_L.  -r  .u~  v~,,,i,-,,rotor  ;u  0  fppt      If.  asain.  we  put  an  —  a,  =  UiJ»  dnu  ^o  — 

depression 


o 
back  fro* the wein  and  diminishes  upwards  ;  but  for  4  feet,  or  complete  cortum  o/  tec*- 

JeSH?  which6  ^CteSLta?  0,5  feet  takes  place,  has  been  found  above  to  be 
1205  feet.  Therefore,  for  each  foot  a  depression  of  ^  feet,  so  that  for  3:  feet,  we 
should  have  °'5  '  3-Z*  =  0,157  feet,  and,  therefore,  the  rise  of  the  back-water  at  2,500 
feet  back  fromTe  weir  is  2  -  0,157  =  1,843  feet,  and,  therefore,  the  depth  of  water 
=  5,843  feet.  If  we  calculate  according  to  the  second  formula 

_  \  UJA  '  *%'    Z,  and  if  we  put  into  this  /=  800,  p0  =  86,  a0  =  1, 

1_2_>V 

b  =  560  r  —  140Q°—  2  5  and  £  =  0075,  we  get  the  depression  corresponding  =  0,399 

£  »d  i'rle  Sn.  PU.  i-W  ^-«W  v-i-W-MM.  -A-««. 

_.  140°  =  2  652  and  £  =  .0075,  the  depression  is  found  to  be  0,383  feet.     Continuing 
I  lhif  «„,„»',  **,  I  «,U  ,ime  =  9W,0  =  85,5,.0  =  6,60,-0,383  =         S 

140°    —288  and  ?  =  00749,  we  get  the  depression  o0—a1 
^  "  ' 


*7U.  jfociw™*-  »W«Z.— If  we  consider  somewhat  closely  the 
equation  of  the  curve  of  the  back-water,  that  is,  of  its  longitudinal 
section,  viz : 

Kr-j^g 

ao-«i  =  I— —2-^- 

\    "«'5   , 

we  discover  several  interesting  circumstances  in  reference  t 
back-water.     In  the  fraction  : 

r     P     v 
sm'a-^F'2~a 


a    % 


BACK-WATER  SWELL. 


147 


the  numerator  and  denominator  become  more  nearly  equal  to  0,  the 
greater  the  velocity  v,  and  according  as  the  one  or  the  other  'first 
becomes  0,  we  have  : 


K— «i).o 


*    ^y 

We  perceive  from  this  that  when  the  numerator  becom«s  =  0,  the 
division  I,  or  the  limit  of  the  back-water  becomes  infinitely  distant, 
and  in  the  case  of  the  denominator  becoming  =  0,  the  length  I  =  0, 
or  there  is  no  back-water.  Now  the  numerator  becomes  =  0,  when 

?  .  2-  .  —  =  sin.  a,  or,  when  the  velocity  of  the  dammed  water  differs 
F     2q 


\-^~ 


F  sin.  a. 


of 


in  an  infinitely  small  degree  from  the  velocity  v 

the  uniformly  flowing  water  of  the  stream,  and  the  denominator  be- 
comes =  0,  when : 

2     v2  v2       a 


that  is,  when  the  height  due  to  the  velocity  =  half  the  depth  of  the 
stream. 

When  the  height  due  to  the  velocity  of  the  water  before  the  intro- 
duction of  a  weir,  is  less  than  half  the  depth  of  the  undammed  water, 
the  back-water  takes  the  form  shown  in  Fig.  143,  and  if  the  height 
due  to  the  velocity  be  greater  than  half  the  depth,  the  back-water  has 
the  form  Fig.  144,  there  being  a  rise  or  swelling  at  the  point  EG. 

Fig.  144. 


If  in  the  equation  sin.  a  =  £  E.  .  ^-,  we  put  ^-  =  -  F  =  ab,  and 

p  (though  it  be  only  approximately)  =  b,  we  have :  sin.  a.  =  %  f . 
Thus  the  circumstances  represented  in  Fig.  144,  are  likely  to  occur 
when  the  fall  or  inclination  of  the  stream  a,  is  greater  than  f  the 
co-efficient  of  resistance  f  =  .0075,  that  is  when  o  >  .00375,  or 
o  >  gs_5  or  i  in  266.  As  rivers  and  water-courses  have  generally 
a  less  inclination  than  this,  the  sudden  depression  EG,  Fig.  144,  is 
eldom  observable  in  them. 

Remark  1.  This  sudden  depression  of  the  back-water  was  first  observed  by  Bidone, 
in  a  12  inch  wide  trough,  in  which  *  was  =0,033.     The  same  appearance  is 
when  the  inclination  of  the  channel  changes,  as  shown  in  the  Fig.  145.     1 
of  inclination  of  the  upper  part  be  greater  than  i  £,  and  inclination  of  the  l< 


148 


RESERVOIRS. 


Fig.  145.  less,  there  is  formed  at  the  point  of  change 

a  swell,  or  sudden  rise  where  the  less  depth 
corresponding  to  the  greater  inclination,  passes 
into  the  greater  depth  corresponding  to  the 
less  inclination. 

Remark  2.  Saint-Guilhem  has  given  an  em- 
pirical equation  for  the  curve  of  the  back- 
water, but    the  author  has  given   one   more 
simple    and    accurate    in    the    "Mlgemeinen 
Maschinen  Encyc.opiidie,"  article  "Bewegung  des  Wassers.'1 

§  71.  Reservoirs. — In  districts  where  the  supply  of  water  is  small, 
but  where  powerful  machines  are  nevertheless  required,  as  in  mining 
districts  generally,  the  construction  of  reservoirs  (Fr.  etangs  ;  Ger. 
Teichen],  or  large  artificial  ponds,  that  fill  during  seasons  of  rain, 
and  supply  the  demands  of  drier  seasons,  is  a  matter  of  practical 
importance.  The  site  to  be  chosen  for  a  reservoir  is  regulated  by  a 
variety  of  circumstances.  The  main  question  is  that  of  the  relative 
level  of  the  machines  to  which  the  water  is  to  be  applied.  This 
being  satisfied,  they  are  most  advantageously  placed  in  a  deep  dean, 
or  part  of  the  valley  where  they  can  collect,  not  only  the  rain-water, 
but  the  streamlets  and  springs  of  as  large  a  surrounding-district  as 
possible.  In  such  a  situation  a  single  dyke  or  dam  going  square 
across  the  valley  is  sufficient  to  enclose  the  reservoir.  The  shorter 
the  dyke,  and  the  less  the  superficial  area  of  a  reservoir  for  a  given 
cubical  contents,  the  better.  The  steeper  the  banks,  therefore,  the 
more  economically  a  reservoir  is  formed.  The  lower  the  level  of 
the  reservoir  compared  to  the  surrounding  district,  the  greater  supply 
of  water  may  be  led  into  it,  or  will  flow  to  it  naturally. 

In  selecting  the  site  for  a  reservoir,  great  attention  must  be  paid 
to  the  nature  of  the  bottom,  that  is,  its  impermeability  must  be 

thoroughly  ascertained;   also  its 
Fig- 14fi-  fitness  for  bearing  the  weight  of 

the  dyke  or  dam.  Artificial  pud- 
dling is,  of  course,  a  resource 
available  in  many  cases;  but  for 
very  extensive  reservoirs,  it  is  a 
precarious  and  expensive  remedy 
for  want  of  natural  impermea- 
bility. Fissures  in  rocks,  depo- 
sits of  sand  and  gravel,  morasses 
or  bogs  are  to  be  avoided  by  all 
means. 

Remark.  On  this  subject,  see  Smeaton's 
"  Reports,"  Sganzin.  "Cowrs  de  Construction,'' 
and  Hagen  "  Wasserbaukunst." 

The  value  of  a  reservoir  depends 
chiefly  on  its  superficial  and  cubi- 
cal contents.      For  ascertaining 
these,  an  accurate  survey  is  neces- 
sary.    The  points  I,  II,  III,  &c.,  of  Fig.  146,  are  laid  down  from  a 


DYKES. 


149 


survey  with  the  chain  or  theodolite,  and  cross  sections  are  then  taken 
by  leveling  (and  sounding,  when  there  exists  a  natural  reservoir) 
on  equi-distant  parallel  lines  0  —  0,  I  —  I,  &c. 

If  60,  b19  62  .  .  .  6n,  he  the  widths  0—0,  I—  I,  II—  II,  &c.,  and 
if  the  distance  between  the  parallelize  a,  the  area  of  the  dam  is: 


and  if,  in  like  manner,  F0,  Fv  F2,  &c.,  be  the  area  of  the  cross  sec- 
tions corresponding  to  the  widths  &0,  bv  62,  &c.,  respectively,  the 
volume  of  the  dam: 

V=[F0+Fn 

By  dividing  the  cross  sections  by  parallel  lines,  drawn  at  equal 
depths,  we  get  the  means  of  laying  down  contour  lines  of  equal 
depth,  and  so  ascertain  the  contents  of  the  dam  for  each  depth. 

Remark.  The  author's  work  "  Der  Ingenieur,"  contains  detailed  instructions  for  measur- 
ing reservoirs  and  dykes. 

§  72.  Dykes.  —  The  dykes  or  dams  of  reservoirs  are  generally  of 
earth-work,  seldom  of  stone.  The  face  inside,  or  next  the  reservoir 
is  covered  with  clay  puddle,  and  with  a  carefully  laid  course  of  gravel. 
They  are  carried  up  of  a  uniform  slope,  or  with  offsets  ir  terraces. 
They  are  carefully  rammed  at  every  foot  of  additional  height  laid 
upon  them.  Especial  care  must  be  taken  with  the  foundation,  which 
must  be  carried  down  to  an  impermeable  stratum  with  which  the 
superstructure  must  be  connected,  so  that  the  bed  of  junction  may 
be  perfectly  water-tight.  When  a  water-tight  substance  cannot  be 
found,  a  system  of  piles  must  be  used  to  insure  this  most  important 
point  of  the  reservoir's  efficiency.  The  depth  of  the  foundations 
depends  on  the  nature  of  the  ground,  as  above  explained,  and  5,  10, 
and  20  feet  deep  foundations  have  been  executed. 

The  dyke,  in  its  main  features,  is  shown  in  Fig.  147,  having  a 
trapezoidal  section  EK 

or  FL.    J1C,  is  sometimes  Fte-  147- 

termed  the  crown  of  the 
dam  ;  it  must  be  well 
paved,  and  generally  has 
a  parapet  wall  to  prevent 
the  wash  of  water  during 
high  winds  from  damag- 
ing the  crown,  or  washing 
over  to  injure  the  back  of 
the  dam  JVE  or  ME.  The 
piece  KME  of  the  dyke 
is  termed  the  middle  or 
centre  piece,  and  the 


pieces  ANH  and  BMC  are  termed  the  wings  of  the  dam.     As  to 
the  dimensions  of  dams,  the  b 
rate  of  1  to  3,  and  the  back 


the  dimensions  of  dams,  the  breast  is  generally  made  to  slope  at  the 
k  at  the  rate  of  1  to  2.     The  width  on 


150 


STABILITY  OF  DYKES. 


the  top  is  very  various.  For  high  dykes  it  varies  from  10  to  20 
feet.  A  common  rule  is,  to  make  the  width  at  top  equal  to  the 
height,  but  this  only  applies  to  dams  of  small  height.  The  dyke 
should  be  carried  from  3  to  6  feet  higher  than  the  highest  water 
intended  to  be  in  the  reservoir. 

Fig.  148  represents  a  cross 

Fig.  148.  section  of  a  dyke  for  a  reser- 

voir. J1BCE  is  the  breast- 
work of  clay  carried  down 
to  water-tight  substratum, 
BGFC  is  the  backing  of 
earth-work,  J1E  is  the  paved 
face,  the  paving  being  4  feet 
thick  at  bottom,  and  2  feet 
at  top. 

Remark.  If  /  be  the  length  along  the  top,  and  /,  the  length  along  the  bottom,  b  the 
breadth  on  top,  and  6,  the  breadth  at  bottom,  and  if  h  be  the  height  of  a  dyke  such  as 
Fig.  148,  the  cubic  contents  of  the  dam  are: 


149- 


In  applying  this  formula,  it  must  be  borne  in  mind,  that  the  well-rammed  clay  does  not 
occupy  quite  one-half  of  that  of  the  earth-work  that  has  not  been  rammed. 

§  73.  Stability  of  Dykes.  —  Dykes  are  exposed  to  the  pressure, 
and  sometimes,  though  rarely,  to  the  shock  or  impetus  of  water. 
They  must,  therefore,  be  of  proportions  that  will  resist  either  being 

overturned  or  shoved  forward  by 
the  action  of  the  water.  The 
conditions  under  which  they  resist 
being  shoved  forward  have  been 
examined,  Vol.  I.  §  280  ;  and  we 
shall  now  consider  the  question  of 
stability  in  reference  to  dislocation 
by  rotation.  The  water  acts  on 
the  internal  slope  or  breast  J1D 
of  a  dyke,  Fig.  149,  with  a  normal 
pressure  OP  =  P,  the  point  of 
application  of  which  is  M  is  at  the 
distance  LM=  f  the  depth  CK=$ 
h  from  the  surface  of  the  water 

(Vol.  I.  §  278).  For  a  length  of  dam  =  1,  P  =  JJD  .  Y  .  ^  y  being 
the  density  of  the  water,  or  weight  of  cubic  unit.  The  horizontal 
component  of  this  pressure  is  :  H  =  h  .  1  .  y  .  -  =  J  h2  y,  and  the 
vertical  component,  (if  m  be  the  relative  batter,  or  mh  the  absolute 
batter  DE  of  the  breast,)  V  =  mh  .  1  .  y  .  -  =  |  mh2?.  The  weight 
of  the  piece  of  the  dyke  of  length  =  1,  acting  at  the  centre  of  gravity 


STABILITY  OF  DYKES.  151 


S  of  the  trapezoidal  section  JIB  CD,  is  G  =  (b  +  m  +  n  h\h7v  in 

which  b  =  the  breadth  AB,  and  n  relative,  or  n  h  the  absolute  batter 
or  slope  of.  the  back  of  the  dyke.  From  P  and  G,  or  from  Jf,  F, 
and  G,  there  arises  a  resultant  force  OR  =  R,  the  statical  moment 
of  which  CJ\T  .  R,  referred  to  the  corner  C,  represents  the  stability 
of  the  dam.  If  we  suppose  P,  and  also  H  and  F,  acting  in  JW,  the 
statical  moment  of  P  =  (statical  moment  of  H  minus  statical  moment 
of  $=  I  h2  y  .  MQ  —  imtfy  .  CQ  =  i  A2  y  (MQ  —  m  .  CQ}=  1 


hence  we  have  the  statical  moment  of  G  working  in  a  contrary 
direction  : 

=  i  n  h2  Vl  .  f  n  h  +  5  £  7l  ^w  A  +  |\  +  m  k2  yx  (n  k  +  5  +  J  m  h) 
=  hVl(%n2  h2  +  nbh  +  %bz  +  %mnh2  +  ±mbh  +  %m*h2) 
=  h  Yl  [(™2  +s2n2+mn^+(n+^bh+ib*J  and,hence, 
we  have  the  stability  of  the  dykes  : 


—  [i  *  —  w  (n,  A  +  J  +  f  w  A)]  -  y  V    In  order  now  to  find  the  point 

eX,  in  which  the  line  -  of  -  resistance  UWX  cuts  the  base  CD  of  the 
dyke,  we  must  determine  the  distance  CX  of  this  point  from  the 

f^v        OJi  7? 

edge  C,  and  for  this  we  put  :  7TT,=  -rr^=  ,,  ,    „  '•>  and  from  this 

CJV         HK         V  -\-  Lr 


F+  G       G+  V 


«'+  3»m)  A'+  (2 
~~ 


By  aid  of  this  formula,  other  points  W  in  the  line  -of-  resistance 
may  be  found,  if  for  h  different  heights  of  dyke  be  introduced,  or 
we  may  ascertain  the  stability  of  any  part  of  the  dam  bounded  by  a 
horizontal  plane. 

For  a  dyke  with  vertical  sides,  m  =  n  =  0,  hence 

a  =  352?i  —  A'y  =  i  b  —  ^JL  (Vol.  II.  §  10).     If  the  inclination 

6  b  Yl  6  b  yl 

of  the  breast  and  back  be  1  to  1,  or  45°,  m  =  n  =  1,  therefore, 
3  (2  A2  +  3  bh  +  £2)  Yl  +  (4  h  +  3  &)  h  y  . 
~ 


and  if  b  =  h,   then  a  =  18  Y*  .      ,  and   if  y,  =  2   y,  then 

4  yx  +  y         3 


152  OFFLET  SLUICES  OF  DYKES. 

a  =  |f  h=  4|  J,  or,  as  in  this  case  the  breadth  at  the  base  b1=  3  6, 
or  b  =  %  blt  a  =  f  ?  6r  According  to  Vauban's  practice,  there  is 

ample  Security  when  a  =  f  .  |-  =  T%  5,  (Vol.  II.  §  11),  so  that  for 

the  last  case  there  is  an  excess  of  stability.  All  things  considered, 
it  is  well  in  dykes,  for  great  reservoirs,  to  make  a  at  least  =  0,4  6,, 
or  the  line  of  resistance  should  cut  the  base  at  T40ths  of  the  width  of 
the  base  from  the  heel  of  the  dyke. 

Example.  Required  the  line  of  resistance  of  a  dyke,  the  batter  of  inclination  of  the 
breast  of  which  m  =  1,  that  of  the  back  n  =  £,  the  breadth  on  the  summit,  or  crown 
)>eing  6=10  feet.  Assuming  that  the  mass  of  the  dyke  has  a  specific  gravity  =  2.  We 

_  2  (3  A3  +60  A  +300)+  (f  A-f30)A_  1200  -f  300^+  17  A*. 

3(3A+40  +  A)  24  (10+  A) 

hence  for  A=0,a  =  5feet;  for  A  =  5feet,a  =  ^  =  8,68  feet;  forA  =  10  feet,a  =  ^ 


=  12,29  feet,  for  h  =  1  5  feet,  a  =      l    =  1  5,87  feet,  for  h  =  20  feet,  a  =  =  1  9,44 

600  i  '20 

feet,  &c.    If  the  height  of  dyke  be  very  great,  we  may  put  :  a    =  -  ,   and   b  =  |  A, 

hence  f-  =  f£.     As  -*J  is  more  than  0,4,  such  a  dam  would  be  safe  for  an  infinite 

b 
height 

Remark.  According  to  the  formula  b  =  —  ^^-  in  the  example  Vol.  I.  §  280,  if  we 

put  a^mh,  then  2  b  =  (3  —  m)  A,  hence  A  ^  -  ,  and,  therefore,  in  our  last  exam- 

3  —  m 
pie,  in  which  m  =  1,  A  =  b  =  10  feet. 

§  74.  Offlet  Sluices  of  Dykes.  —  Offlet  sluices  and  discharge-  pipes, 
or  culverts,  must  be  provided  in  the  reservoir  dyke.  The  offlet  sluice 
or  regulator,  serves  for  the  discharge  of  any  excess  of  water  that 
would  accumulate  in  times  of  extraordinary  wet.  The  discharge- 
pipe  or  culvert,  is  for  supplying  the  lead  or  water-course  as  circum- 
stances require.  There  may  be  one  or  more  of  each  of  these  acces- 
sories in  a  dyke.  For  instance,  in  some  dykes  an  offlet  is  arranged 
at  the  very  lowest  level,  so  that  the  dam  may  be  completely  emptied 

when  occasion  requires,  and  above 
this,  a  second  offlet  is  laid,  by  which 
the  water-course  is  supplied  with 
water  to  be  led  to  the  machine  that 
is  to  receive  it  as  power. 

The  offlet-pipes  may  be  either  of 
wood  or  iron,  or  of  stone,  or  may  be 
built  culverts.  Fig.  150,  in  the  mar- 
gin, gives  a  general  idea  of  the  ar- 
rangement of  the  drawing-sluice  or 
discharge-sluice  of  a  dyke.  Ji  is 
the  end  of  the  pipe  or  culvert,  on 
the  face  of  which  is  a  flat  piece  of 
wood  or  iron  B,  CD  is  a  cast  iron 
or  wooden  sluice-board,  fitting  into 


WATER-COURSES.  153 

guides,  DE  is  the  sluice-rod,  reaching  to  the  surface  or  top  of  the 
dyke,  E  is  a  cross  piece  by  which,  in  the  absence  of  grooves  or 
guides  on  the  plate  on  the  end  of  the  pipe,  the  sluice  is  kept  pressed 
upon  its  bed,  G  is  a  strong  beam  having  a  female  screw,  through 
which  the  screw  GH  passes,  and  the  handle  or  key,  H,  being  turned, 
the  screw  elevates  or  depresses  the  sluice-rod,  as  may  be  desired, 
for  opening  and  shutting  the  sluice. 

The  discharge-pipe  must  have  a  sectional  area,  such  that  the  dis- 
charge when  the  water,  or  rather  its  head,  is  lowest,  may  be  sufficient 
for  the  supply  of  the  power  required  for  the  machine.  If  Q  be  the 
quantity  of  water  to  be  discharged  per  second,  h  the  given  least  head, 
I  the  length,  and  d  the  diameter  of  the  discharge-pipe,  ?  the  co-efficient 
of  resistance  at  entrance,  and  f  ,  the  co-efficient  for  internal  friction, 
then,  according  to  Vol.  I.  §  332, 


or,  more  simply: 


(1  +  0  d  +  ^l     /4Q 


d  =  0,4787 

If,  therefore,  we  take  £  from  the  table  in  Vol.  I.  §  325,  and  ?,  from 
the  table  in  Vol.  I.  §  331,  we  can  determine  by  approximation  the 
required  width  of  pipe.  As  the  head  is  higher,  a  greater  part  of 
the  aperture  must  be  closed,  so  that,  according  to  Vol.  I.  §  338, 
there  must  be  introduced  a  greater  co-efficient  of  resistance  for  the 
entrance.  If  the  entrance  aperture  be  very  small,  the  water  does 
not  fill  the  pipe,  and,  therefore,  the  calculation  is  simply  referable 

to  the  area  of  the  opening  F  = ?  where  /*  is  to  be  taken 

n  \/2  g  h 

from  Vol.  I.  §  325.  With  table  of  areas  of  segments,  the  calcula- 
tions are  very  simple.  The  prolongation  of  the  discharge-pipe 
through  the  dyke  must  be  of  very  substantial  cement-built  masonry, 
and  in  large  dykes  should  be  from  5  to  6  feet  high. 

Example  1.  A  discharge-pipe  of  100  feet  long  is  required  to  let  off  10  cubic  feet  per 
second,  when  the  head  is  reduced  to  1  foot,  what  must  be  the  diameter'?  Supposing  the 
inclination  of  the  sluice  to  be  40°  (equal  that  of  the  breast  of  the  dyke)  then  £  =  0,87; 
and  the  co-efficient  £,  corresponding  to  a  velocity  of  5  feet  =  0,022,  we  have  rf=4787 
^(1,870  rf-j-  2,2)  .  100,  and  d=  1,7  satisfies  this  equation  very  nearly.  Thus,  a  dis- 
charge pipe  of  1,7  .  12  =  20,4  inches  would  fulfil  the  required  conditions. 

Example  2.  In  what  position  must  this  sluice-board  be  placed,  in  order  to  discharge 
only  10  cubic  feet  of  water  per  second,  when  the  head  is  16  feet?  If  we  assume  that 
the  pipe  does  not  fill  in  this  case,  then 

F  =         Q         — 10        _  =     5     =.431  square  feet. 

0,731  .  802  v/lti          n>6 


This  segment  of  radius  —  reduced  to  radius  I  =  0,431 =  0,598,  and   from    a 

2  2,89 

table  of  areas  of  segments,  we  find  the  height  of  such  a  segment  to  be  5  inches. 

§  75.  Water-courses. — The  water  of  the  reservoir  is  conducted  or 
led  to  the  point  at  which  it  is  to  be  applied,  i.  e.,  to  the  machine 
through  which  it  is  to  expend  its  mechanical  effect,  by  canals,  water- 


154 


WATER-COURSES. 


courses,  and  mill-leads.  These  channels  are  generally  dug  out  of 
the  natural  soil,  raised  upon  embankments  and  aqueducts  over  the 
deeper  valleys,  and  cut  as  drifts  or  tunnels  through  the  greater  ele- 
vations that 'occur  in  their  course.  The  bed  of  the  canals  are  formed 
of  sand  or  gravel,  on  a  bottom  of  clay,  or  are  hand-laid  stones,  or 
concrete  formed  with  cement,  and  not  unfrequently  it  consists  of  a 
wooden,  an  iron,  or  a  stone  trough.  The  sides  of  this  canal  form 
right  lines,  or  its  section  is  a  gently  curved  trapezium,  or  it  is  rec- 
tangular when  it  becomes  a  trough.  The  section  of  water-courses 
is  from  1J  to  3  times  as  wide  as  its  depth.  The  slopes  of  the  sides 
of  the  course  are  generally  very  slight,  or  none  at  all  in  the  case  of 
masonry  set  in  cement.  An  inclination  of  1  in  2  is  given  to  dry 
stone  sides,  an  inclination  of  1  in  1  in  the  case  of  compact  earth  or 
clay,  and  of  2  to  1  in  the  case  of  sand  or  loose  earth.  Fig.  151 
gives  an  idea  of  the  construction  of  a  water-course  in  loose  ground, 


Fig.  151. 


Fig.  152. 


not  water  tight.  Fig.  152  represents  the  manner  of  forming  such 
a  course  on  the  side  of  a  hill,  where  the  earth  taken  from  the  cut 
is  made  the  supporting  bank  on  the  under  side.  Fig.  153  shows  the 


Fig.  153. 


Fig.  154. 


manner  in  which  it  is  sometimes  necessary  to  construct  the  embank- 
ments for  aqueducts. 

Fig.  154  is  a  section  of  a  walled  drift  or  tunnel,  through  ground 
not  considered  impermeable  to  water,  and  incapable  of  standing 
unsupported.  The  manner  of  putting  troughs  together  is  indicated 


WATER-COURSES. 


155 


in  the  sketches  in  Fig.  155  for  wood,  and  Fig.  156  for  iron,  where 
the  flanges,  bolted  together,  are  further  made  water-tight  by  what  is 


Fig.  155. 


Fig.  156. 


termed  a  rust-joint  (a  cement  composed  of  sal  ammoniac  and  iron 
filings  or  turnings). 

The  junction  of  a  water-course  with  a  river  JlJl,  Fig.  157,  should 
be  gradually  widened  and  rounded  off,  and  the  head  D  substantially 
finished,  so  that  it  may  not  be  injured  by  freshes,  or  objects  carried 
against  it  in  time  of  floods.  Flood-gates  or  sluices  have  to  be  ar- 
ranged along  the  course,  if  this  be  of  any  considerable  extent.  These 
sluices  should  be  made  self-acting,  that  no  damage  may  be  done  to 


Fig.  157. 


Fig.  158. 


the  banks  by  even  a  momentary  overflow  (the  self-acting  sluices  on 
Shaw's  water-works,  in  Scotland,  is  the  most  notable  case  of  this 
self-acting  arrangement  on  record).  They  act  generally  by  a  float 
being  raised  as  the  water  in  the  channel  rises,  which  float  opens  a 
valve  or  sluice  to  discharge  the  surplus  water  in  convenient  localities. 
Sometimes  a  case  fills  as  the  water  rises,  overcomes  a  counter- 
balance, and  in  its  descent  opens  a  valve  or  sluice,  by  which  the 
surplus  water  is  discharged.  The  syphon,  properly  adapted,  forms 
a  simple  contrivance,  and  is  shown  in  Fig.  158,  where  JiBC  is  the 
syphon  with  an  air-pipe  DE.  When  the  water  in  the  water-course 
rises  to  the  height  of  the  summit  of  the  syphon,  which  is  the  highest 
point  for  safety,  the  syphon  fills  with  water,  and  the  water  is  drawn 
off  and  discharged  at  C,  the  head  being  CH,  the  depth  of  C  under 
the  water  surface.  When  the  water  has  sunk  to  the  level  of  DE, 


156  WATER-COURSES. 

the  air  rushes  in  and  stops  the  action  of  the  syphon.  If  the  water 
does  not  fill  the  section  BD  of  the  pipe,  the  discharge  is  made  under 
the  conditions  of  a  weir. 

§  76.  The  velocity  of  the  water  in  a  water-course  should  be  neither 
too  slow,  for  then  the  course  chokes  with  weeds;  nor  too  fast,  for  then 
the  bed  of  the  channel  may  be  disturbed ;  and  besides,  too  much  fall 
must  not  be  lost  in  the  inclination  of  the  course. 

A  velocity  of  7  to  8  inches  per  second  is  necessary  to  prevent 
deposit  of  slime  and  growth  of  weeds,  and  1£  feet  per  second  is  ne- 
cessary to  prevent  deposit  of  sand.  The  maximum  velocity  of  water 
in  canals  depends  on  the  nature  of  the  channel's  bed. 

On  a  slimy  bed  the  velocity  should  not  exceed    £  foot, 
clay 

1     « 


sandy 
gravelly 
shingle 
conglomerate 
hard  stone 


2 

4 

5 

10 


This  applies  to  the  mean  velocity. 

From  the  assumed  mean  velocity  (?,  and  the  quantity  of  water  to 
be  led  through  the  course  Q,  we  have  the  section  F,  and  hence  the 
perimeter  p  of  the  water  section.  If  we  put  this  in  the  formula 

5  =  -  =  £  .  P.  .  —  (Vol.  I.  §  367),  we  get  the  required  inclination 

8  of  the  canal,  and  hence  the  fall  required  for  the  lead,  whose  length 
=  Us  h  =  8  I 

The  inclination  may,  therefore,  be  very  different  according  to  cir- 
cumstances. As,  however,  ?  as  a  mean  is  0,007565,  and  c  generally 

from  1  to  5  feet,  and  -^  is  something  between  \  and  2,  the  limits  of 

» 
the  inclinations  for  the  water-course  would  be 

0,007565  .  \  .  1  .  ,0155  =  0,000023,  and 

0,007565  .  2  .  25  .  ,0155  =  0,00578, 

the  courses  leading  from  the  machine  have  a  greater  fall,  that  the 
machine  may  be  quite  clear  of  back-water.  The  course  leading  from 
the  machine  is  usually  termed  the  tail-race. 

Remark  1.  The  water-courses  for  the  water  wheels  and  general  uses  of  the  Freyberg 
mining  districts,  have  inclinations  varying  from  >  =  0,000'25  to  J  =  0.0005,  or  from  15 
inches  to  30  inches  per  mile,  the  tail  races  generally  .001  to  .002.  The  Roman 
aqueduct,  at  Arcueil,  near  Paris,  has  an  inclination  >  =  0,0004  1  6,  or  2  feet  per  mile 
nearly.  The  New  River,  which  supplies  a  great  part  of  London,  has  an  inclination 
J=  0,00004735.  [The  Croton  aqueduct  has  J  =  0,000208,  or  1,1  foot:  and  the  Boston 
aqueduct  0.000047j5,  or  3  inches  per  mile,  the  same  as  New  River.]—  -Ai«.  ED. 

Remark  2.  All  sudden  changes  of  sectional  area  and  of  direction  are  to  be  avoided, 
because  these  not  only  occasion  loss  of  fall,  but  entail  other  bad  effects  in  the  way  of 
wear  and  tear  and  deposits.  Bends  or  curves  should  have  as  great  a  radius  as  possible, 
or  the  sectional  area  should  be  increased  there.  If  r  be  the  mean  width  of  the  course,  and 
R  the  radius  of  curvature,  the  fall  lost  by  a  curve  may  be  calculated,  according  to  Vol.  I. 
§  334,  by  the  formula: 


until  we  have  further  experimental  data. 


SLUICES.  157 

Remark  3.  The  deposit  of  slime,  sand,  and  the  growth  of  plants,  diminishes  the  section 
of  water-courses,  and  fall  is  thereby  lost.  The  water-courses  must,  therefore,  be  carefully 
cleaned  out  from  time  to  time. 

§  77.  Sluices.  —  The  entrance  of  water  into  a  water-course  is 
either  free,  or  regulated  by  a  sluice.  If  the  water  enter  unimpeded 
from  the  weir-darn  or  reservoir,  in  which  it  may  be  considered  to  be 
still,  the  surface  of  the  water  sinks  where  the  flow  commences,  and 
the  depression  is  proportional  to  the  initial  velocity  in  the  water- 

course, and  therefore  =  —  ,  which  height  must  be  deducted  from 

the  total  fall  of  the  water-course.     For  moderate  velocities  of  3  to  4 
feet  per  second,  this  depression  amounts  to  only  1J  to  3  inches. 

If  .the  entrance  of  water  into  the  lead  be  regulated  by  a  sluice, 
two  distinct  cases  may  present  themselves.  Either  the  water  flows 
freely  through  the  sluice,  or  it  flows 
into  and  against  the  water  of  the  _  Fig.  159. 
lead.  It  will  generally  be  found 
that  the  depth  of  the  water  in  the 
lead,  is  greater  than  the  height  of 
the  sluice-opening,  and,  therefore, 
there  occurs  a  sudden  rise  S  at  a 
certain  distance  from  the  sluice  AC, 
Fig.  159.  The  height  BC  =  x  of 
this  rise  is  a  function  of  the  velo- 
city v  of  the  water  in  the  lead,  and  of  the  velocity  vl  of  the  water 
coming  up  to  the  sluice,  such  that 

x  —  ^  --  —  ,  and  if  we  deduct  this  height  from  that  due  to  the 
2<7       2# 

velocity  vlt  or  jiC=  h  =  J-,  then  the  head  causing  the  initial  velo- 
city v  is  : 


or  exactly  the  same  as  if  the  water  were  discharging  freely.  As  the 
sluice-opening  is  never  perfectly  smooth,  there  is,  of  course,  a  certain 
resistance  increasing  the  head  required  by  10,  or  even  more,  per 
cent. 

If  we  put  G  =  the  area  of  the  section  of  the  water  flowing  in  the 
lead,  and  F=  the  area  of  the  sluice-opening  CD,  then  Gv  =  Fvv 
and,  therefore,  the  rise 

.  *  =  a-a, 

and  substituting  for  J-  the  height  due  to  the  velocity  or  the  head 
AC  =  A,  x  =  f~l  —  (-)2~1  h.  If  the  difference  x  =  a  —  a,  of  the 

depth  of  water  a  and  atbe  less  than  fl  —  (^Vl  £-»  the  rise  occurs 
VOL.  ii.  —  14 


158 


PIPES,   CONDUIT  PIPES. 


Fig.  100. 


further  down  the  lead;  but,  if  it  be  greater,  then  the  rise  occurs 
nearer  the  sluice,  till  at  last  the  dis- 
charge takes  place  under  back-water, 
as  shown  in  Fig.  160.  In  this  case, 
the  head  JlB  =  h  has  not  only  to  pro- 
duce the  velocity  v  in  the  water  of 
the  lead,  but  also  to  overcome  the  re- 
sistance arising  from  the  sudden 
change  of  the  velocity  v}  into  the  velo- 
city v  of  the  lead.  If  we  put  F=  the 
area  of  the  opening,  and  G  =  the  area 

of  the  lead,  the  loss  of  head  occasioned  by  this  transition  is: 


and  hence  the  fall: 


.'.It  is  obvious  that  the  difference  of  level  of  the  water  before  and 
behind  the  sluice,  is  so  much  the  greater,  the  smaller  the  sluice- 
opening  F  in  proportion  to  the  section  of  the  water  in  the  lead  G. 

Example.  A  lead  of  5  feet  mean  width,  and  3  feet  depth,  supplies  45  cubic  feet  per 
second.  It  is  fed  through  a  sluice  4  feet  wide,  and  1  foot  opening.  Required  bow  much 
higher  the  water  will  stand,  before  the  sluice  than  behind  it.  G  =  5  X  3  =15  square 
feet,  f  =  4  X  1  =  4  square  feet ;  v  =  \  f  =  3  feet  per  second,  and  v ,  =  — '—  =  4/ 
=  lli  feet. 

Now  as  f  1  —  (^Y  ~j  —  =  [1  —  (T45y]  2»02  =  J.88  feet  is  les9  than  a  —  a,  =  3  —  1 
=  2  feet,  it  is  evident  that  there  will  not  be  a  free  discharge.     The  formula 
A  =  I  1  -f-  ( l)      —  gives  the  difference  of  level  required 

A  =  (1  -f-  2,75")  0,139  ==  8,56  X  0,139  =  1,19  feet,  which  must,  however,  be  increased 
10  per  cent  at  least,  on  account  of  the  resistances  at  the  opening. 

§  78.  Pipes,  Conduit  Pipes. — Pipes  are  usually  employed  when 
smaller  quantities  of  water  are  to  be  brought  to  supply  machines, 
such  as  the  water-pressure  engine,  and  turbines  of  very  high  fall. 
They  have  the  advantage  of  much  greater  pliability  than  open  con- 
duits, but  their  adoption  instead  of  open  canals,  depends  entirely  on 
local  circumstances  in  the  question  of  relative  advantage. 

Pipes  are  made  of  wood,  of  pottery,  of  stone,  of  glass,  iron,  lead, 
&c.  Wooden  and  iron  pipes  are  those  most  usually  employed  in 
connection  with  water-power  engines.  Wooden  pipes  are  usually 
formed  from  large  trees,  because  straight  pipes  of  12  to  20  feet  in 
length,  and  from  1J  to  8  inches  bore,  or  internal  diameter,  may  be 
got  from  this  timber.  The  bore  is  generally  ^  of  the  diameter  of 
the  tree.  Wooden  pipes  are  jointed  or  connected  together  as  shown 
in  Figs.  161  and  162. 


PIPES,  CONDUIT  PIPES. 


159 


Fig.  161  is  a  conical  mortice  with  a  binding  ring  and  packing  of 
hemp,  or  linen  steeped  in  tar  and  oil.     Fig.  162  is  a  connection  by 


Fig.  161. 


Fig.  162. 


means  of  an  iron  double  spigot  going  from  1  to  2  inches  into  the 
ends  of  the  two  pipes. 

Iron  pipes  are  the  most  durable  and  most  universally  employed 
of  all  pipes.  They  are  cast  of  any  diameter,  and  have  been  used 
as  large  as  5  feet  bore.  The  length  of  each  pipe  rarely  exceeds  12 
feet,  and  is  less  as  the  diameter  is  greater.  For  3  feet  diameter, 
they  are  about  9  feet  long  each,  in  England.  To  prevent  internal 
oxidation,  they  are  sometimes  boiled  in  oil,  sometimes  lined  with 
wood,  or  with  Roman  cement.  The  thickness  of  metal  must  be  pro- 
portional to  the  pressure  they  have  to  bear,  and  to  their  diameter, 
according  to  Vol.  I.  §  283.  The  jointing  of  iron  pipes  is  effected 
either  by  flanges  and  bolts,  as  shown  in  Fig.  163,  there  being  an 
annular  packing  between  the  flanges,  or  by  the  spigot  and  faucet,  as 
shown  in  Fig.  164,  (which  is  considered  the  best  and  cheapest  mode, 


Fig.  163. 


Fig.  164. 


when  the  packing  is  properly  done  with  small  folding  wedges  of 
hard  wood.)  A  collar,  or  ring,  as  shown  in  Fig.  165,  is  sometimes 
used.  The  packing  is  either  leather,  felt,  lead, 
iron  rust,  or  wood.  The  more  effectually  to 
prevent  all  leakage,  there  is  sometimes  a  small 
internal  ring  put  in  (counter-sunk)  to  cover 
the  joint.  A  flexible  joint,  as  shown  in  Fig. 
166,  is  sometimes  necessary  (as  for  crossing  a 
river,  where  it  is  necessary  to  let  the  pipe  rest 
on  the  original  bed  of  the  river).  Where  the  pipes  are  exposed  to 
changes  of  temperature,  expansion  joints,  as  shown  in  Fig.  167, 


Fig.  166. 


Fig.  167. 


1GO  PIPES,   CONDUIT  PIPES. 

must  be  introduced,  that  the  expansion  and  contraction  of  each  con- 
siderable length  may  not  injure  the  pipes  or  joints.  The  expansion 
of  cast  iron  is  .0000111  of  its  length,  for  each  degree  of  centigrade ; 
and,  therefore,  for  a  change  of  temperature  of  50°,  or  from  winter 
frost  to  summer  heat,  the  expansion  would  be  0,000553.  Therefore, 
for  every  900  feet,  there  is  an  expansion  and  contraction  of  6  inches. 
This  is  to  be  compensated  by  an  arrangement,  such  as  is  shown  in 
our  last  figure,  where  the  pipe  B  is  movable  through  the  water-tight 
stuffing  box  C.  There  should  be  a  compensation  joint  for  every 
length  of  300  feet  exposed  to  a  change  of  temperature. 

§  79.  Pipes  cannot  of  course  be  laid  so  as  to  maintain  a  straight 
line ;  but  rise  and  fall,  and  turn  from  right  to  left  in  their  course. 
It  is  a  general  maxim  to  avoid  all  sudden  changes  of  direction  in 
laying  pipes.  All  bends  should  be  effected  by  curved  pipes,  of  as 
great  radius  as  possible,  or  the  bore  of  the  pipe  should  even  be 
increased  at  bends,  to  avoid  loss  of  vis  viva  in 
168.  the  water.  When  a  pipe  bends  over  an  eleva- 

tion, as  in  Fig.  168,  there  is  a  disadvantage 
arises  from  the  collection  of  air  at  Z,,  as  this 
contracts  the  section,  and  would  gradually  stop 
the  flow  of  water.    To  prevent  this  accumulation 
of  air,  vertical  pipes  AL,  called  ventilators  or 
windpipes,  are  placed  on  the  summit  of  the 
pipe,  through  which  air,  or  other  gases  given  off 
by  the  water,  can  be  discharged  from  time  to 
time,  by  means  of  a  cock,  to  be  turned  by  the 
inspector  of  the  pipes.     To  make  these  ventilators  self-acting,  the 
arrangement  shown  in  Fig.  169  has  been  adopted.    In  this  ventilator 
the  discharge  valve  V  is  connected  with  a  float  S 
Fig.  169.  of  tinned  iron,  which  is  pressed  upwards  as  long 

as  it  is  surrounded  by  water,  and  thus  keeps  the 
valve  shut,  but  falls  or  sinks  downwards  when  the 
space  about  it  becomes  filled  with  air,  and  then  the 
valve  is  opened  to  discharge  the  air.  As  air  col- 
lects at  the  highest  points  of  a  conduit  pipe,  so  the 
sand  or  slime  collects  at  the  lowest  points.  To 
remove  any  deposits  of  this  nature,  waste-cocks  are 
placed  at  these  points,  by  which  the  pipe  is  scoured, 
or  separate  receptacles  for  the  deposits  are  attached 
to  the  pipes,  and  these  are  cleared  from  time  to 
time,  as  may  be  found  necessary.  The  deposit  is 
favored  by  the  greater  section  of  these  receptacles,  and  sometimes 
by  the  introduction  of  check  or  division  plates,  which  still  more  retard 
the  flow. 

Cocks  for  flushing  the  pipes  are  introduced  more  or  less  frequently, 
according  to  the  purity  of  the  water,  and  the  rate  of  flow  through 
the  pipes,  and  seldom  at  less  intervals  than  100  feet.  For  ascer- 
taining the  point  in  the  pipe  where  any  obstruction  has  occurred, 
piezometers  (Vol.  I.  §  344)  are  very  useful. 


PIPES,   CONDUIT  PIPES.  161 

For  regulating  the  discharge  of  water  through  pipes,  cocks  and 
slides,  and  valves  are  used.  The  effect  of  these  has  been  shown  in 
Vol.  I.  §  340,  &c.  In  order  to  moderate  the  effects  of  the  impulse 
or  shock  arising  on  the  sudden  closing  of  a  cock,  or  other  valve,  it 
is  useful  to  have  a  loaded  safety  valve,  so  placed  that  it  will  open 
outwards  when  the  pressure  exceeds  a  certain  limit. 

Remark.  The  most  detailed  treatise  on  the  subject  of  conduit  pipes,  is  Geniey's  "Essai 
sur  les  Moyens  de  conduire,  deleter,  et  de  distributer  les  eaux."  Matthew's  "  Hydratdia,"  and 
the  "  Civil  Engineer  and  Architect's  Journal,''  contain  much  useful  information  on  this  sub- 
ject Hagen,  "  Wasserbaukunst,"  Vol.  I.  has  a  chapter  on  water  pipes. 

§  80.  The  general  conditions  of  motion  in  conduit  pipes  have  been 
already  discussed.  If  h  be  the  fall,  and  I  the  length,  d  the  diameter 
of  the  pipe,  £  the  co-efficient  of  resistance  at  entrance,  ^  the  co- 
efficient for  friction  in  the  pipe,  and  £2,  &c.,  the  co-efficients  for 
resistances  in  passing  bends,  cocks,  &c.,  and  if  v  be  the  velocity  of 
discharge,  we  have : 


and  if  Q  be  the  quantity  of  water : 


We  see  from  this,  that  for  carrying  a  certain  quantity  of  water  Q, 
so  much  less  fall  is  requisite,  the  greater  the  width  of  the  lead.     If 
'there  be  two  pipes  instead  of  one,  the  two  together  having  an  area 
equal  to  the  one,  and  supposing  each  to  take  half  the  whole  quantity  , 
of  water,  the  fall  necessary  is  : 


.)(!«)• 

/  \  *  / 


d  ./  J 


so  that  in  this  case  the  fall  is  greater,  or  the  head  required  is  greater, 
so  that  it  is  mechanically  better  to  employ  one  large  pipe,  than  two 
smaller  of  equal  section  when  united. 

Calculations  for  whole  systems  of  pipes,  where  there  are  numerous 
subdivisions  of  branches,  become  exceedingly  complicated.  The 
case  in  which  water  is  brought  from  different  sources,  and  the  pipes 
ultimately  united,  is  of  the  same  nature. 

The  general  nature  of  such  calculations  is  as  follows.  If  the  sub- 
division takes  place  in  a  reservoir  which  has  a  much  greater  sectional 
area  than  the  main  pipe,  the  water  comes  there  again  to  rest,  or  the 
whole  vis  viva  is  destroyed,  and  has  to  be  acquired  again  in  the 
branch  pipes.  The  same  loss  of  vis  viva  occurs  when  several  branches 
come  together  in  a  reservoir,  from  which  one  main  pipe  carries  off 
their  waters.  In  this  case,  the  calculation  reduces  itself  to  a  sepa- 
rate consideration  of  each  branch  and  pipe,  and  requires  no  further 
elucidation.  The  collecting  reservoir  should  be,  if  possible,  placed 
at  such  levels  as  will  ensure  the  same  mean  velocity  in  all  the  pipes, 
in  order  that  the  loss  of  head  or  of  vis  viva  may  be  the  least  possible. 

*14 


162 


PIPES,   CONDUIT  PIPES. 


In  the  case  of  a  simple  subdivision  or  fork,  it  is  mechanically  ad- 
vantageous to  make  such  arrangements  that  the  water  may  move  in 
all  the  pipes  with  the  same  velocity.  If,  besides  this,  the  branches 
be  curved  off  properly,  so  that  there  is  no  sudden  change  of  direc- 
tion in  the  passage  of  the  water  from  the  main  into  the  branches,  it 
may  be  assumed  that  there  is  no  loss  of  head  or  vis  viva.  In  the 
case  sketched  in  Fig.  170,  let  h  =  the  head  BC,  I  the  length,  and 

Fig.  170. 


d  the  diameter  of  the  main  pipe,  and  let  At  =  the  head  or  fall  D^EV 
ll  the  length,  and  d1  the  diameter  of  the  one  branch,  and  d2=  D2E2, 
Z2,  and  c?2,  the  fall,  length,  and  diameter  of  the  other  branch,  and 
also  let  <?,  cr  <?2,  be  the  velocities  of  the  water  in  these  three  branches, 
and,  lastly,  let  C  be  the  co-efficient  of  resistance  for  entrance,  and  ^ 
the  co-efficient  for  friction  of  the  water.  Then,  for  the  length  of 
pipes  JJCEV  we  may  put : 


for  the  length  of  pipes 


But  the  quantity  of  water  Q  =  - —  c  of  the  main  pipe,  is  equal  to 


d? 


of  the   two 


the  sum  of  the  quantities  Q1=  .      '  cv  and  Q2  =  _ 
branches;  and  hence  we  may  put: 

By  aid  of  these  three  equations,  three  quantities  may  be  deter- 
mined. The  more  usual  case  is,  that  of  the  fall,  the  length  and  the 
quantity  of  water  being  given,  the  necessary  diameter  of  the  pipe  is 
required.  If,  then,  we  assume  a  certain  velocity  c  in  the  main,  we 
get  the  width  of  this  pipe  by  the  formula: 

d  =     I ,  and  we  have  then  only  to  solve  the  equations: 


PIPES,  CONDUIT  PIPES.  163 

By  transformation,  we  get  similar  equations  for  determining  d  and 
</2,  as  in  Vol.  I.  §332,  viz.: 


we  can,  therefore,  as  in  Vol.  I.  §  332,  put: 


\ 


and  in  order  to  obtain  a  first  approximation  to  the  values  of  dt  and 
cl2,  we  may  omit  these  from  the  part  under  the  radical.  If  ca  and  c2 
come  out  to  be  very  different  from  c,  attention  must  be  paid  to  the 
co-efficient  fr  being  variable,  and  its  value  for  each  of  the  pipes  in- 
troduced, and  the  determination  of  dl  and  d2  repeated. 

Example.  A  system  of  pipes,  to  consist  of  one  main  and  two  branches  is  intended  to 
carry  15  cubic  feet  of  water  per  minute  by  one  branch,  and  24  cubic  feet  by  the  other. 
The  levels  showed  that  in  a  length  of  1000  feet  of  main,  the  fall  was  4  feet,  the  first 
branch  had  a  fall  of  3  feet  in  600  feet  length,  and  the  second  1  foot  in  200  feet.  What 
must  be  the  diameters  of  the  pipes  respectively?  If  we  suppose  a  velocity  of  2£  feet 
per  second  in  the  main,  then  its  diameter 


=    /!?  =    /  4  •  39  —    I_^L  =  0,57  54  feet  =  6,9  inches.     If  now  (according  to 

-s/wc        Wf.60«      W25*- 

ol.  I.  §  436),  we  put  the  co-efficient  of  resistance  for  ent 
f  friction  (Vol.  I.  §  435)  for  velocity  c  =  2,5  feet,  £  ,  =  0 

_  )2=  1,621,  we  have  for  the  diameter  of  the  branches 
.»/ 


Vol.  I.  §  436),  we  put  the  co-efficient  of  resistance  for  entrance  £=0,505,  the  coefficient 
of  friction  (Vol.  I.  §  435)  for  velocity  c  =  2,5  feet,  £  ,  =  0,0253,  and  as  2  g  =  64,4,  and 


64.4  .  7  — (0,505  +  0,0253  .  1738)  . 


5 (0,0253.  200 -M,       6        ("I—'/5'06*7'-     If  we  first  neglect  rf,  and  rf2 
""•J    322,0  —  277,98  U<sJ        *J     169,7 

under  the  radical,  we  get  the  approximate  values  rf,  =    I— '—  =0,39  feet,  and 

>/  1706 

_  s  /5,06  _  Q  495  feet     jf  we  now  introciuce  the  value  on  the  right-hand  side  of  the 
2       «J  169,7 

equation,  we  get  more  accurately  dt  =    I.  '.  '      =  0.391  feet  and 

</,  =  J|  5'555— OfiOfi  feet     The  diameter  rf,  =  0,391  corresponds  to  a  velocity 
J  169,7 

4  1 


.  .  __ 

*d?       0,39  1'. 
and  the  diameter  rfa  =  0,505  corresponds  to 

ca=  |J  ___  -  -  =  1,997  feet, 

6  J  j 


2,082  feet, 


104  WATER  POWER. 

and  hence  we  should  have  more  accurately  for  the  first  branch  pipe  £,  =0.0263,  and  for 
the  other  &=  0,0270,  and  hence  with  the  best  accuracy  which  the  formula  admits 
</,  —  J/0,0263  .600+  U,391  _  5ll6,171    _  0)394  feet  =  4,7  inches,  and 

>/  1706  *j    1706    ~ 

</2  =  4  /0.0270  .  200  -f  0,505  _  *  /  5,905  =Q^n  feet  =  6>13  inches. 

>/  169,7  W  169,7 


CHAPTER   IV. 

OF    VERTICAL    WATER    WHEELS. 

§  81.  Water  Power.  —  Water  acts  as  a  moving  power,  or  moves 
machines  either  by  its  weight,  or  by  its  vis  viva,  and  in  the  latter 
case  it  may  act  either  by  pressure  or  by  impact.  In  the  action  of 
water  by  its  weight,  ifris  supported  on  some  surface  connected  with 
the  machine,  that  sinks  under  the  weight  ;  and  in  the  action  by  its 
vis  viva  it  comes  against  a  surface  yielding  to  it,  in  a  horizontal 
direction  generally,  which  is,  in  like  manner,  an  integral  part  of  the 
machine.  If  Q  be  the  quantity  of  water  (or  Q  y  the  weight  of 
water)  available  as  power,  per  second,  and  h1  the  fall^  or  the  per- 
pendicular height  through  which  the  water  falls  in  giving  out  its 
mechanical  effect,  then  the  mechanical  effect  produced  is  :  L  —  Qy  . 
h  =  Q  h  y.  If,  again,  c  be  the  velocity  with  which  the  water 
comes  upon  any  machine,  the  mechanical  effect  produced  by  its  vis 
viva,  is  : 


That  water  may  pass  from  rest  to  the  velocity  c,  a  fall,  or  height 
due  to  the  velocity  h  =—  is  necessary,  and,  therefore,  in  the  second 

instance  we  may  also  put  L  =  h  Q  y.  So  that  the  mechanical  effect 
inherent  in  water  is  the  product  of  its  weight  into  the  height  from 
which  it  falls,  as  in  the  case  of  other  bodies. 

Water  sometimes  acts  by  its  weight  and  vis  viva  simultaneously, 
by  combining  the  effects  of  an  acquired  velocity  c,  with  the  fall  h 
through  which  it  sinks  on  the  machine.  In  this  case,  the  mechanical 
effect  produced  is  again  : 


The  mechanical  effect  Pv  yielded  by  a  machine  is  of  course 
always  less  than  the  above  available  mechanical  effect  Q  h  y  ;  be- 
cause many  losses  occur.  In  the  first  place,  all  the  water  cannot 
always  be  brought  to  work  ;  secondly,  a  part  of  the  fall  is  generally 
lost  ;  thirdly,  the  water  retains  a  certain  amount  of  vis  viva  after 
having  quitted  the  machine  ;  and,  fourthly,  there  are  the  passive 


WATER  WHEELS.  —  BUCKET  WHEELS.  165 

resistances  of  friction,  &c.,  interfering.     The  efficiency  of  a  water- 
power  machine  may  be  represented  by  M  =  JLlL*  and  the  merits  of 


different  machines  are  proportional  to  the  approximation  of  this 
ratio  in  their  case,  to  unity. 

From  the  general  formula  L  =  Q  h  y,  it  is  manifest  that  fall  and 
quantity  of  water  are  convertible  terms  ;  so  that,  by  doubling  the 
height  of  a  fall  with  a  given  quantity  of  water,  we  have  the  same 
power  as  by  doubling  the  quantity  of  water,  and  retaining  the  ori- 
ginal height. 

Example.  There  is  a  fall  of  10  feet  yielding  12  cubic  feet  of  water  per  second.  The 
machine  uses  only  8,5  feet,  however,  and  the  water  leaves  it  with  a  velocity  of  9  feet 
per  second,  and  the  friction  is  ascertained  to  be  750  feet  Ibs.;  required  the  efficiency  of 
tliis  machine. 

The  available  mechanical  effect  i  =  12  X  10  X  62,5  =  7500  feet  Ibs.  (Pruss.),  and  the 
effect  of  the  fall  used  =  12  X  8,5  X  623  =  6375  feet  Ibs.  The  mechanical  effect  lost  from 
the  vis  viva  retained  in  the  water  leaving  the  machine  is  0,0165  X  9*  X  '2  X  62,5  =  941,2 
feet  Ibs.  ;  and  the  mechanical  effect  consumed  by  friction  =  750  feet  Ibs.;  and,  therefore, 
the  useful  effect  of  this  machine  Pv  =  6375—  (941,2  -f  750)  =4083.8  feet  Ibs.,  and  the 

efficiency  =^^=.624. 
7500 

§  82.  Water  Wheels.  —  The  machines  used  as  recipients  of  water- 
power,  are  either  wheels,  (water  wheels,  Fr.  roues  hydrauliques  ; 
Ger.  Wasserrader  ;)  or  engines  with  pistons,  water-pressure  engines, 
(Fr.  machines  d  colonnes  d'eau;  Ger.  Wassersa'ulen-maschinen.) 
Water  wheels  are  essentially  "the  wheel  and  axle,"  with  water  as 
power.  Pressure  engines  consist  of  a  column  of  water,  pressing  on 
a  movable  piston. 

Water  wheels  are  either  vertical,  the  axle  of  the  wheel  being  hori- 
zontal, or  they  are  horizontal,  the  axle  of  the  wheel  being  vertical. 

Vertical  water  wheels,  concerning  which  we  shall  first  treat,  are 
either  overshot,  (Fr.  roues  en  dessus;  Ger.  Oberschlagige,)  or  breast 
wheels,  (Fr.  roues  de  cote;  Ger.  Mittelschlagige,)  or  undershot,  (Fr. 
roues  en  dessus;  Ger.  Unterschlagige.)  The  water  comes  on  to  the 
wheel  near  the  top  or  summit,  in  overshot  wheels  ;  near  the  middle 
or  level  of  the  axle,  in  breast  ;  and  near  the  bottom  in  undershot 
wheels.  In  the  first,  the  water's  weight  is  chiefly  the  source  of 
mechanical  effect,  whilst  in  undershot  wheels  it  is  the  inertia  of  the 
water,  and  in  breast  wheels,  the  weight  and  inertia  both  that  are 
usually  effective.  Undershot  wheels  sometimes  hang  freely  between 
boats  in  a  wide  stream,  and  sometimes  in  a  confined  course,  which^ 
is  either  straight  or  curved.  Breast  wheels  are  generally  hung  in  a 
curved  channel  or  course.  It  is,  perhaps,  necessary  to  distinguish 
from  the  above-named  vertical  wheels,  Poncelet's  wheel,  in  which  the 
water  acts  by  pressure  in  its  ascent  and  descent  on  curved  buckets. 

§  83.  Bucket  Wheels.  —  All  vertical  water  wheels  consist  of  an 
axle  of  wood  or  iron,  with  two  journals  or  gudgeons  —  of  two  or  more     ,  +\ 
annular  crowns  or  shroudings  —  of  a  set  of  arms  connecting  the 
shrouding  with  the  axle,  and  of  a  series  of  cells  or  buckets  between 
the  shrouding  —  and,  lastly,  of  a  flooring,  which  reaching  from  crown 


166  CONSTRUCTION  OF  WATER  WHEELS. 

to  crown  on  their  under  side,  forms  a  close  cylinder.  The  buckets 
divide  the  annular  space  bounded  by  the  shroudings  on  the  flooring 
into  a  series  of  compartments,  which,  when  the  buckets  are  placed 
more  tangentially  than  radially,  form  water  troughs  or  cells.  This 
latter  is  the  general  construction  of  the  buckets  of  overshot  and  breast 
wheels,  which  are  thus  distinct  from  the  simple  floats  of  undershot 
wheels.  For  overshot  wheels,  the  water  is  led  on  to  the  wheel  by 
a  trough  or  channel  having  a  regulating  sluice,  and  falls  thence  into 
the  second  or  third  cell  from  the  summit  of  the  wheel.  If,  then, 
the  wheel  be  once  in  motion,  each  cell  gets  partially  filled  with 
water  as  it  passes  the  discharge  of  the  water  trough  or  lead,  and 
retains  the  water  till  near  to  the  bottom  of  the  wheel,  when  it  falls 
out,  so  that  there  is  always  a  certain  number  of  cells  filled  with 
water  on  one  side  of  the  wheel,  and  this  keeps  the  wheel  continuously 
revolving.  Overshot  wheels  have  been  constructed  for  falls  varying 
from  8  to  50  feet,  and  sometimes  even  up  to  64  feet  in  height,  and 
for  quantities  of  water  varying  in  every  degree  up  to  50  cubic  feet 

[of  water  per  second.  It  is  often  more  advantageous  to  put  up  two 
or  three  smaller  wheels,  than  one  very  large  one ;  for  the  weight  of 
the  parts  becomes  inconvenient. 

The  fall  of  a  water  wheel  should  be  measured  as  between  the  sur- 
face of  the  water  at  the  pentrough,  or  regulating  sluice,  and  the  sur- 
face of  water  in  the  tail  race,  the  depth  of  which  latter  will  depend 
of  course  on  the  quantity  of  water,  and  on  the  breadth,  and  the 
inclination  of  the  race.  In  order  to  lose  as  little  of  the  effect  as 
possible,  the  bottom  of  the  wheel  should  be  as  near  as  possible  to 
the  surface  of  the  race,  so  that  the  height  from  the  surface  of  water 
in  the  pentrough  to  the  bottom  of  the  wheel  may  also  serve  as  a 
true  measure  of  the  height  of  fall.  If  there  be  any  risk  of  back- 
water in  the  race,  the  wheel  must  be  hung  at  an  extra  elevation  ac- 
cordingly. 

§  84.  Construction  of  Water  Wheels. — Water  wheels  are  made  of 
wood  or  of  iron,  or  of  both  these  materials  combined.  The  manner 
of  uniting  the  axle  and  arms  together  is  various.  In  the  case  of 
wooden  wheels,  they  are  either  strapped  or  bolted  on  to  the  side  of 
a  square  axle,  as  shown  in  Fig.  171,  or  they  are  let  into  the  axle  by 
morticing,  or  passed  through  it.  The  latter  construction  is  bad, 
and  only  applicable  to  light  wheels.  The  arms  of  the  framed  wheel, 
Fig.  171,  may  be  strengthened  by  braces  or  auxiliary  arms.  Such 
wheels  of  20  to  50  feet  diameter  are  erected  for  pumping  water,  for 
driving  ore  mills,  &c.,  in  the  Freiberg  mining  district.  A  is  the 
axle,  B  and  C  are  the  journals  or  gudgeons,  DE,  FG,  &c.,  are  the 
main  arms,  HM,  HL  are  the  auxiliary  arms  ;  DFG,  and  D1F1G1  are 
the  shroudings*of  the  wheel ;  K  is  the  pentrough  end.  The  crowns* 
are  two  rings  of  wood  composed  of  8  to  16  pieces  of  3  to  5  inch-thick 
segments.  The  whole  is  put  together  with  screw  bolts.  There  are 
cross  tie-bolts  for  uniting  the  two  crowns.  The  interior  of  the  crowns 
are  grooved  out  to  receive  the  buckets.  The  open  wheel  JV  is  a  part 
of  the  mechanism  for  transmitting  the  motion. 


CONSTRUCTION  OF  WATER  WHEELS. 
Fig.  171. 


167 


Fig.  172  is  an  iron  water  wheel.  Cast  iron  discs,  or  naves  BD, 
are  set  on  the  axle  AC,  and  to  these  the  arms  are  attached  by  bolts. 
An  intermediate  ring  or  crown  is  introduced  when  the  wheel  becomes 
more  than  7  or  8  feet  wide,  and  this  has  either  a  separate  set  of 
arms  or  diagonal  arms,  as  shown  by  -BG,  &c.,  brought  from  this  to 
the  nave  of  the  outer  crowns.  Through-bolts  are  introduced  to  bind 
the  whole  firmly  together.  The  prime  mover  in  the  train  of  me- 
chanism is  often,  as  shown  in  Fig.  172,  a  toothed  wheel,  forming 
the  periphery  of  an  outside  crown  ELF,  and  this  works  into  a  pinion 
on  a  lying  shaft  MN.  In  practice,  this  pinion  should  be  rather 
below  than  above  the  level  of  the  axle,  and  on  the  side  on  which  the 
water  is.  The  buckets  are  of  sheet  iron,  and  bolted  to  ribs  of 
angle  iron,  cast  on  the  inner  surface  of  the  crowns,  or  fastened  to 
them. 

§  85.  Dimensions  of  Parts. — The  axle,  the  gudgeons,  and  the 
arms  of  the  wheel,  must  have  dimensions  proportioned  to  the  weight 
and  power  of  the  wheel.  To  find  these,  the  principles  and  rules  of 
the  third  section  of  the  first  volume  are  to  be  applied.  The  dimen- 
sions of  the  axle  may  be  determined  either  in  reference  to  the  mo- 
ment of  inertia  of  the  wheel  and  the  resistance  to  torsion  of  the  axle, 


168 


/*-*>     t 

DIMENSIONS  OF  PARTS. 

Fig.  172. 


or  in  reference  to  the  weight  of  the  wheel,  and  the  resistance  of  the 
axle  to  transverse  strain.  In  Vol.  I.  §  211,  it  has  been  shown,  that 
in  the  case  of  a  solid  round  cast  iron  axle  of  radius  =  r,  acted  upon 
by  the  statical  moment  of  an  effort  P  equal  to  Pa,  that  Pa  =  12600 

r3,  where  r  and  a  are  expressed  in  inches.     Hence  r  =     I  Ofir)f. 

inches  =  the  radius  of  axle;  and  if  a  be  expressed  in  feet,  then  the 
diameter  of  the  axle 

3      .  12  Pa        3\4~Pa       n  1ft.    *  5-  .     , 
12600-  =  JW  =  0,197  v/Pa  inches. 


But  the  mechanical  effect  corresponding  to  the  moment  Pa,  u,  being 

P.  «ua 

30  .550' 


±sut  the  mechanical  enect  corresponding  to  the  momen 
the  number  of  revolutions  of  the  wheel  per  minute,  is 

L  =  Pv  =  P 


rtu  a  r.    ,  ,,  .,          ,  T        P  .  nu  a 

teet  IDS.,  or,   in  horses    power,  L  — 


hence  Pa 


30 
16500  L 


ft  U 


,  and 


j 


16500 


^       niQT  3  \L       Q  0/1  3IL  .    . 

d  =  0,197  -  .      —  =  3,34    f—  inches. 

\      it         \u  \ju 

But  for  greater  secuvitv,  we  generally  make  d  —  6,12      f—  inches. 

\ju 


DIMENSIONS  OF  PARTS.  169 

If  the  axle  be  square,  the  side  of  the  square 

5  =  3  \J*!L. .  d  =  0,94  d,  i.  e.,  5  =  5,75  *  \-  inches. 
NSv/2  V« 

If  the  axle  be  made  hollow,  the  formulas  given  in  Vol.  I.  §  209 
and  §  210  are  to  be  used  with  the  above  co-efficients.  Wooden 
axles  should  be  from  3  to  4  times  as  large  in  diameter  as  iron  axles. 

If  the  toothed  wheel,  transmitting  the  power  of  the  water  wheel, 
be  an  integral  part  of  it,  as  in  Fig.  172,  the  axle  undergoes  a  less 
torsion-strain  by  the  moment  of  the  power,  and,  therefore,  its  dimen- 
sions should  be  determined  in  reference  to  the  weight  of  the  wheel. 
For  this  we  may  make  use  of  the  formulas  given  in  Vol.  I.  §  202, 

Q  {SJa  —  - \  =  ^ bh2,  in  which  we  substitute  for  Q,  G  the  weight 

\  /         8/6 

of  the  water  wheel,  c  the  breadth  of  the  wheel,  /  the  length  of  the 
axle,  and  lt  and  12  the  distance  of  the  centre  of  the  wheel  from  the 
two  gudgeons.  Hence  for  a  square  axle: 


And  if  for  —  we  put  1000  Ibs.  as  a  minimum,  and  expressing  I, 

t> 
and  /2,  and  c  in  feet,  we  get  for  square  cast  iron  axles: 

s  =  0,229  3  I G  (*jJ*  —  -}  inches, 

\     V  I         8/ 
and,  on  the  other  hand,  for  round  cast  iron  axles: 


Wooden  axles  must  be  made  at  least  as  large  again. 

The  diameter  of  the  gudgeon  dl  is  deduced  from  the  well-known 

formula  given  in  Vol.  I.  §  196,  PI  =  -  r3  K,  substituting  in  it  for 

/j  .  i_  0--t>  . 
r  =  -£  ,  and  I  the  length  of  the  gudgeon,  which  is  generally  about 

equal  to  d..  its  diameter.     Hence  we  should  have  for  the  diameter 

.-*£  -  <^.  <?*$-_ 

d.=    I——  .  P,  for  which  we  may  put  in  practice  d,  ="*fc£f:K/P, 

\*  K 
P  being  the  pressure  on  the  gudgeon.     Buchanan's  rule  is  dl  = 


0,241  v 

The  arms  of  the  wheel  must  evidently  be  of  strength  sufficient  to 
resist  the  moment  of  rotation.  If  this  moment  be  again  taken  =  Pa, 
and  the  number  of  the  arms  in  each  set  of  arms  of  the  wheel  =  n, 
so  that  for  a  double  set  of  arms  the  total  number  of  arms  =  2  n, 

Pa 
then  the  moment  which  a  single  arm  has  to  resist  =  ^-  •     •">  n(>w» 

b  =  the  breadth,  and  h  the  thickness  of  an  arm,  and  if  the  length 
of  the  arm  be  equal  to  the  radius  of  the  wheel  =  a,  then,  from  Vol. 

VOL.  II.—  15 


1TO 


AXLES  AND  GUDGEOXS. 


I.  §  196,  we  have  —  =  6A2  — ,  or,  as  b  is  made  =  mh,  or,  in  iron 
2n  6 

generally,  $  A,  and  in  wood,  f  h,  i.  e.,  ^  =  mh3  _,  and  hence  the 
thickness  of  the  arms  sought,  measured  in  the  direction  of  the  plane 

of  revolution,  is:  h  =  3  I- — -•      If  we  introduce  the  effect,  and 

^  mnK 
number  of  revolutions  of  the  wheel,  then,  for  cast  iron  arms,   h 


10,4 


,4  3  I—  inches.     And,  as  the  diameter  of  the  axle  was  found 

^1  7J|£ 

=  6,12  3  l^l,  we  have  also  h  =  -^',  or  -  =  _l^  ,    and,    there- 


fore, for  4,  6,  8,  10,  12,  16  arms,  the  values  of  - 


1,08,  0,94, 


0,85,  0,79,  0,75,  0,67,  and  from  h,  we  deduce  the  breadth  6,  mea- 
sured in  the  direction  of  the  axis. 


For  wooden  arms  h  = 
=  1  h. 


— ,    and   hence   we   can   deduce 


According  to  Rettenbacher,  the  number  of  arms  in  a  set,  or  to 
one  crown  (of  which  there  are  always  two  at  least),  is  n  =  2  /-  4-  1  j. 

If  a  wheel  be  8  feet  wide,  or  wider,  the  number  of  sets  of  arms 
should  not  be  less  than  three. 

Example.  A  cast  iron  water  wheel,  weighing  35.000  Ibs,  gives  an  effect  of  60  horse- 
power. making  4  revolutions  per  minute:  required,  the  dimensions  of  its  principal  parts. 

The  diameter  of  a  solid  a 


xle  is  rf=6,12     L^=  13,2  inches,  and  that  of  its  gudgeons 

r5UU'J  —  6£   inches,  which  might  be   made  7   inches.     Buchanan's  for- 
inches.    For  the  arms,  supposing  two  sets  of 


</,  =  0,048 

mula  gives   rf,  =  0,241    v  17 SOU 

12  each,  the  thickness  h 


1.7  X  13,2 


— III.  ^  10  inches  nearly,  and  the  breadth  b 


10 


=  2  inches  (h  being  in  the  direction  of  the  plane  of  revolution). 

§  86.  Axles  and  Gudgeons. — We  must  make  special  allusion  to 
the  manner  of  putting  the  gudgeons  in  the  axles,  and  to  the  plum- 
tner  blocks  on  which  they  rest.  For  wooden  axles,  oak,  or  larch,  or 
beech,  answers  exceedingly  well.  They  are  dressed  into  polygons, 
when  the  arms  are  to  be  framed  on  the  axle,  and  they  are  squared 
when  the  arms  are  to  be  morticed  through,  or  into  the  axle.  The 

Fig.  173. 


AXLES  AND  GUDGEONS. 


171 


gudgeons  are  either  spiked  in,  as  shown  at  Z,  Fig.  173,  or  they  are 
hooped,  as  shown  in  Fig.  174.  Also,  plate  or  flat  ends  are  used,  as 
in  Fig.  175  (and  these  are  the  most  common),  or  rings,  as  at  Fig. 
176,  or  compound  gudgeons,  as  at  Fig.  177.  To  strengthen  the 


Fig.  175. 


Fig.  176. 


Fig.  177. 


neck  of  the  axle,  to  prevent  its  splitting,  it  is  dressed  off  conically, 
and  three  iron  rings,  i  to  J  inch  thick,  and  1|  to  3  inches  broad, 
are  driven  on  while  hot.  The  plates  in  the  flat  gudgeon  ends  are 
from  1  to  3  inches  thick,  and  about  an  inch  narrower  than  the 
diameter  of  the  axle.  The  ring  attachment  is  convenient,  when  a 
spur  wheel  is  to  be  placed  at  the  neck  of  the  axle ;  the  compound 
gudgeon  is  applied  when  much  wear  is  anticipated,  because  the  end 
plates  are  easily  removed  and  renewed.  Cast  iron  axles  are  either 
hollow  or  solid,  either  round  or  polygonal  in  section,  and  sometimes 
ribbed  or  feathered  to  increase  their  stiffness. 

For  solid  axles  the  gudgeon  is  generally  in  one  piece  with  the 
'axle.     Fig.  178  is  a  simple  round  axle,  Fig.  179  is  a  feathered  axle, 


Fig.  178. 


Fijr.  ISO. 


Fig.  179. 


and  Fig.  180  is  the  end  of  a  hollow  iron  axle,  with  a  gudgeon  put 
in,  and  an  arm  plate  or  nave  set  upon  it. 

The  gudgeons  rest  on  supports  termed  plummers  or  plumbing 
blocks,  which,  to  afford  a  permanent  seat  for  the  wheel,  are  placed 
on  substantially  founded  walls.  The  plumbing  block  is  lined  with 
a  brass  or  other  movable  seat  for  the  gudgeon.  These  seats  are 
either  of  brass  (hence  termed  generally  brasses),  or  of  gun-metal  (8 
parts  copper,  1  part  tin),  or  of  white  metal ;  sometimes  they  are  of 
wood,  though  seldom. 

The  gudgeons  rest  on  a  wooden  block  in  Fig.  171.     Fig.  181  is  a 


172 


THE  PROPORTIONS  OF  WATER  WHEELS. 


simple,  uncovered,  cast  iron  block.     Fig.  182  is  an  open  block,  with 
a  metal  seat,  or  lining,  and  Fig.  183  is  a  close  or  covered  block  with 


Fig.  181. 


Fig.  182. 


Fig.  183. 


Fig.  184. 


metallic  lining.  The  plumbing  blocks  are  bolted  down  by  means  of 
bolts  and  sole  plates  to  the  walls  or  beams  on  which  the  wheel  is  to 
rest.  The  cover  of  blocks  is  always  provided  with  a  hole,  through 
which  grease  can  be  supplied.  The  inside  of  the  cover  is  sometimes 
grooved,  so  that  the  grease  diffuses  more  readily  over  the  gudgeon. 
And,  wherever  it  is  desired  to  reduce  the  resistance  from  friction  to 
a  minimum,  a  grease  cup,  affording  a  constant  supply,  is  placed  in 
communication  with  a  hole  in  the  plumbing-block  cover. 

§  87.  The  Proportions  of  Water  Wheels.  —  The  first  or  main  ele- 
ment of  a  water  wheel  is  the  velocity  of  the  circumference  v,  or  the 
number  of  revolutions  u.  It  will  be  seen  in  the  sequel,  that  over- 

shot wheels  should  have  a 
very  small  velocity.  Many 
wheels  have  a  velocity  of  10 
feet  per  second,  but  5  feet  is 
more  suitable,  yet  under  2| 
feet  is  not  advisable.  The 
velocity  c  of  the  water  enter- 
ing the  wheel,  should  depend 
on  the  velocity  of  the  wheel, 
and  is  either  equal  to  this, 
or  greater  in  a  certain  pro- 
portion. For  creating  the 
velocity  c,  a  fall  or  height  of 
head,  JIB  (Fig.  184)  =  h, 

c*    .  , 

=  —   is  necessary,  leaving 

of  the  total  fall  AF=  h,  only 
the  fall  on  the  wheel  =  BF 


As  even  in  the  case  of  the  most  perfect  discharge,  6  per  cent,  of  vis 
viva  is  lost,  it  is  advisable  to  take  it  as  10  per  cent,  in  this  case, 
and,  therefore,  to  put  the  effective  fall  required  to  bring  the  water 

on  to  the  wheel  with  suitable  velocity  h1  =  1,1  .  —  ,  and  hence  h2 

c2 
=  h  —  1,1  .  —  .  From  the  fall  on  the  wheel  A2,  we  deduce  the 

2# 
semi-diameter  of  the  wheel  CF  =  CS  =  a,  by  assuming  the  angle 


THE  PROPORTIONS  OF  WATER  WHEELS.  173 

SCD  =  e,  by  which  the  point  of  entrance  of  the  water  D  deviates 
from  the  summit  S  as  given. 

Then  h2  =  CF  +  CB  =  a  +  a  cos.  &  =  (1  +  cos.  0)  0,  and  hence, 
inversely,  a  =  _  —  —  Ai  .     From  the  radius  of  the  wheel  a,  and  the 

1  -j_  COS.  ® 

velocity  y  at  the  circumference,    the  number  of  revolutions  per 

minute  u  =  —  -. 
tta 

When  w  is  given,  we  can  determine  a  and  v.     As  v  =  *ua,  and 

30 

itua          ,.  ,  .  .    c         , 

c=  x  -_  _,  m  which  x  is  a  given  ratio  -,  we  have: 
oO  v 


and  hence  a  =  -  1  --  *  ^^  ^e  solution  of  this  quad- 

1  +  cos.  & 
ratic  equation  gives  : 


-,       _  x/0,000772  (*  u)2  A  4.  (1  +  cos.  0)2-—  (1  +  cos.  &} 

0,000386  (»  W)2  ~'ai 

hence: 

2.  „  =  ^f=  0,1047.  ua. 
oU 

Example  1.  For  a  fall  of  30  feet,  a  wheel  is  to  be  constructed  to  have  8  feet  velocity 
at  circumference,  and  taking  on  the  water,  at  12°  from  the  summit  with  twice  the  above 
velocity.  What  is  the  radius  of  wheel  required,  and  what  the  number  of  revolutions? 
c  =  2  X  8  =  16  feet,  and  hence  Al==  1,1  X  0,0155  X  162  =  4,36  feet,  and 


1  +  cos.  12°        1,978  *  X  12,9 

Example.  2.  If,  inversely,  the  number  of  revolutions  be  5,  then  for  the  above  fall,  and 
other  proportions  *  =  2,  and  the  radius  of  the  wheel : 

_  ^2,316 +  3.9125  — 1.978  _  0,5177  _  13  41 

O0386  ~~  0,0386 

Again,  the  velocity  at  the  circumference  v  =  0,1047  X  5  X  13,41  =  7,02  feet,  the  velo- 
city at  entrance  =  14,04  feet,  and  lastly,  the  height  of  fall  due  to  this  latter  velocity  =  At 
=  1,1  +  0,0155  X  14,04*  =  3,47  feet. 

§  88.  The  proportions  of  the  wheel,  in  reference  to  depth  of  the 
shrouding7  and  width  of  the  wheel,  are  important.  The  depth  of  the 
crown*  (or  water  space)  is  made  10  to  12  inches,  and  sometimes  even 
14  to  15  inches,  and  this  proportion  is  chosen,  because  the  water  in 
a  wheel  with  shallow  shrouding,  acts  with  greater  leverage  than  it 
would  do  on  a  wheel  of  equal  radius  with  deeper  crowns.  As  to  the 
width  or  breadth  of  the  wheel,  it  depends  on  the  capacity  to  be  given 
to  the  wheel.  If  d  be  the  depth  of  crowns,  and  e  the  width  of  the 
wheel  (or  distance  between  the  internal  surfaces  of  the  crowns),  then 
the  section  of  the  annular  space  above  the  flooring  of  the  wheel  is 
=  d  e,  and  if  v  be  the  velocity  at  the  middle  of  the  crown's  depth, 
the  capacity  presented  to  the  water,  per  second,  is  d  e  .  v.  But 
this  cannot  be  considered  equal  to  the  quantity  of  water  delivered 

15* 


174  FORM  OF  BUCKETS. 

on  the  wheel,  because  a  certain  portion  of  this  capacity  is  taken  up 
by  the  substance  of  the  buckets,  and  it  is  also  inexpedient  to  fill  up 
the  buckets  to  the  brim.  We  must,  therefore,  put  d  e  v  =  t  Q,  in 
which  equation  t  >  1  ;  *  is  usually  =  3  to  5,  the  former  when  the 
buckets  are  filled  rather  in  excess,  the  latter  when  they  are  deficiently 
filled.  The  width  of  wheel  is,  however,  now  determined: 

t  Q  it  a  U     ,  30  *  Q          Q  rr     *   Q 

e  =  _-*,  or  as  v  =  -^-,  hence  e  =  -  -5  =  9,5o  —  :* 
dv  30  rtuad  uad 

and  taking  *  =  4,  then  e  =  38,2          .     That  wheels  of  very  great 

u  d  d 

diameter  may  not  be  too  narrow,  it  is  advisable  to  assume  t  =  5. 

The  number  of  buckets  n  is  another  important  element  in  the 
construction  of  water  wheels.     The  more  cells  there  are,  the  longer 
will  the  water  be  retained  on  the  wheel.     But  this  number  has  its 
limits,  because  the  buckets  occupy  space,  taken  from  the  capacity 
of  the  wheel,  and  the  more  the  capacity  is  diminished  for  a  given 
quantity  of  water  delivered  on  the  wheel,  the  sooner  the  water  will 
leave  it.     As  iron,  that  is  sheet  iron  buckets,  are  much  thinner  than 
those  of  wood,  we  may  adopt  a  greater  number  of  iron  buckets,  than 
we  should  do  of  wooden  buckets.     We  may  fol- 
Fig.  185.  low  the  rule,  to  place  the  buckets  at  such  a 

distance  from  each  other,  that  at  that  point 
where  the  wheel  begins  to  spill,  or  lose  its  water, 
the  bucket  next  above,  JIB  D,  Fig.  185,  shall  not 
dip  into  the  water  of  the  one  below  at  B,  for  if 
we  put  the  buckets  closer  than  this,  the  upper 
bucket  diminishes  the  capacity  of  that  under  it, 
and  so  what  we  gain  in  one  respect  is  lost  in 
another.  The  number  of  buckets  is  generally 
made  n  =  5  a  to  6  a,  or  according  to  Langsdorf 
n  =  18  -H  3  a  ;  in  which  expressions  a  is  the  radius  of  the  wheel  in 

feet:  or  the  distance  between  any  two  buckets  is  made  =  7  (l  -f  —  \ 

inches.  From  the  given,  or  thus  found  number  of  buckets  n,  we 
have  the  angle  of  subdivision  0,  i.  e.,  the  central  angle  between  two 

O£*  AO 

adjacent  buckets,  3=  -  . 
n 

Example   Suppose  an  overshot  wheel  of  15  feet  radius,  having  1  foot  depth  of  crown 
and  taking  10  cubic  feet  of  water  per  second,  makes  5  revolutions  per  minute,  the  witLh 

of  the  wheel  must  be  38,2        1Q       =  5,1  feet,  and  the  distance  between  two  buckets  is 
0  .  1  D  .  1 

to  be  7  (  i  -|-  I£")  =  y  ==  1  5,4  inches,  and  hence  the  number  of  buckets  =  2-*-15-12 

\         lu'  15,4 

=  73.  or  72  for  the  sake  of  easier  division  of  the  circle.     The  angle  of  subdivision  is 


§  89.  Form  of  Buckets.  —  The  form  of  the  cells  or  buckets  is  of 
much  consequence  to  the  efficiency  of  water  wheels.  The  buckets 
must  have  such  form  and  position,  that  the  water  may  enter  freely, 


FORM  OF  BUCKETS. 


175 


Fig.  186. 


remain  in  them  to  as  near  the  hottom  of  the  wheel  as  possible,  hut 
no  further.  By  the  various  forms  adopted,  these  requirements  are 
more  or  less  perfectly  fulfilled.  The  two  requirements  are  in  fact 
often,  to  a  certain  extent,  incompatible  ;  for  if  the  cells  be  made  very 
close,  the  entrance,  as  well  as  the  exit  of  the  water,  becomes  much 
impeded.  If  the  buckets  be  merely  plane-boards,  as  shown  at  AD, 
Fig.  186,  the  entrance  of  the  water  is  quite  free  certainly,  but  then 
it  leaves'  the  cells  too  soon,  so  that  there  is  a 
great  loss  of  mechanical  effect.  To  prevent 
this  too  early  loss  of  water,  the  bucket  would 
have  to  be  very  long,  and  the  angle  ADE,  at 
which  the  bucket  inclines  to  the  radius  CE, 
very  large,  i.  e.,  nearly  a  right  angle.  As  this 
is  a  practical  difficulty  in  construction,  it  is  pre- 
ferred to  make  the  bucket  in  two  parts,  or  by 
a  second  piece  DB,  to  give  the  bucket  a  bottom 
or  flooring  of  its  own.  The  bottom  DB  is 
sometimes  termed  the  start,  or  shoulder,  and 
the  outer  piece  BA,  the  arm,  or  wrist.  The 
former  is  generally  placed  in  the  direction  of  the  radius,  sometimes 
at  right  angles  to  the  outer  piece,  or  arm."  The  circle  passing 
through  the  elbow  B,  made  by  the  junction  of  the  shoulder  and  arm, 
is  termed  the  division  circle.  In  the  older  construction  of  wheels, 
-this  circle  is  generally  found  placed  at  £  of  the  depth  of  the  shrouding 
from  the  interior,  or  the  sole  of  the  wheel.  As,  however,  the  capacity 
of  a  cell  is  greater  the  wider  the  shoulder-blade  DB  (Fig.  187)  is,  or 
the  greater  the  angle  JIBE  (which  we  term  the  elbow  angle],  we  now 
usually  find  the  division  circle  in  the  middle  of  the  depth  of  the 
shrouding.  The  capacity  of  a  cell  will  then  depend  only  on  the 
width  or  position  of  the  arm.  The  simplest  construction  of  buckets, 
is  to  make  the  end  A  of  the  arm  JIB  start  from  the  prolongation  of 


Fig.  187. 


Fig.  188. 


the  shoulder  next  above  it  D.B,,  or,  by  letting  the  arm  be  included 

360° 

between  the  sides  of  the  division  angle  fl  == But   this   con- 

N 

struction  does  not  close  or  cover  the  cells  sufficiently,  except  for  very 
shallow  shrouding,  and,  therefore,  the  usual  plan,  for  wheels  up  to  35 
to  40  feet  diameter,  is  to  let  the  arm  extend  over  f  of  the  dimension 
angle,  or  the  arc  EA  is  made  =  f  EE,,  Fig.  188.  From  the  radius 


17t>  FORM  OF  BUCKETS. 

CA  =  0,  and  the  central  angle  JICB  =  $„  included  by  the  arm,  we 
can  easily  find  the  elbow  angle  ABE  =6.     In  the  triangle  *iCB  as 

CB=  CE—BE=a  —  b-,  then: 
tang. 


b       a(l-cos.^) 
^—  * 

Example.  A  30  feet  wheel,  shrouding  10  inches  deep,  is  to  have  (according  to  Langs- 
dorf's  rule)  18+  3  X  15=63  buckets,  or,  say  64.  and  each  is  to  extend  over  |  of  the 
division  angle.  What  will  be  the  elbow  angle?  We  have: 

6  =  »  «J>  =  £*  °,  hence  8  =  *  X  V  =  1  J*°  =  7°    l/>  °2"  5'  and 
j  _  '  5  *'n'  7°'  1''  5v!"'  5       _  —       3o."  0,1  2241        _  4,40678 

tang.     =  -s~3TiD  (1  -co*.  7°,T',  52",  5)  "~  1  —  36  X  0,00752  ~  0,72928' 
hence  J=fcU°,  3ti'. 

§  90.  The  position  of  the  arm  of  the  bucket  may  likewise  be  de- 
termined, by  adopting  as  a  rule,  that  the  least  section  of  a  cell  shall 
be  somewhat  greater  than  the  section  of  the  water  coming  on  to  the 
wheel.  If  the  cells  come  exactly  under  the  water-.;Vtf,  then  this  con- 
struction would  permit  the  water  to  enter  freely,  and  the  air  to 
escape  unhindered.  This  may  be  done  as  follows.  From  a  point  of 
division  .B,,  Fig.  189,  in  the  division  circle  (or 
_  Fig-  1S9-  _  line  of  pitch  of  the  buckets)  with  a  radius  greater 
by  one  or  two  inches  than  the  thickness  of  the 
water-jet,  or  layer,  describe  a  circle,  and  from 
the  next  adjacent  point  of  division  .B,  draw  a 
tangent  Bjl  to  this  circle.  This  is  the  position 
of  the  arm  of  the  bucket  required,  for  then  the 
least  width  -B,JV*  of  the  cell  ^BD1  is  equal  to 
the  radius  of  that  circle.  But  in  order  to  de- 
termine the  thickness  of  the  layer  of  water 
coming  (from  the  pentrough)  on  to  the  wheel, 
we  must  know  the  fall  At  from  the  surface  of  the  water  in  the  pen- 
trough  to  the  point  J^JV";  also  the  quantity  of  water,  and  the  width 
e  of  the  wheel.  As  Q  =  rf,  e  V'2ghv  therefore,  the  thickness  in 

question  d1  =  -         ,  and  to  allow  of  free  escape  of  air,  we  make 
Ax 

+  1  or  2  inches.     If  equal  spaces  for  the  exit 


of  air  and  entrance  of  water  be  allowed,  then  d,  = 

e 
be  the  equation  to  be  satisfied. 

Buckets  in  three  parts,  as  AEBD,  Fig.  190,  give,  cseteris  paribus, 
more  capacity  than  those  in  two  parts,  without  any  greater  contrac- 
tion. There  is,  therefore,  mechanical  advantage  in  this  form,  though 
it  be  more  expensive  to  execute.  Curved  buckets  are  best  of  all,  as 
shown  in  Fig.  191,  and  this  is  the  great  advantage  of  sheet  iron,  to 
which  this  form  can  readily  be  given.  If  the  section  of  bucket  is  to 


SLUICES. 


177 


be  composed  of  two  segments  of  circles,  then  it  is  only  necessary  to 
find  the  position  of  the  arm  of  a  bucket,  by  any  of  the  planes  above 


Fig.  190. 


Fig.  191. 


given — at  its  bisection  M  (Fig.  191),  to  erect  a  perpendicular,  and 
from  any  point  0  at  will  to  describe  an  arc  with  the  radius  OB,  and 
from  any  point  K  in  it,  to  describe  another  arc  to  complete  the  bucket 
AED  of  a  suitable  form. 

Example.  If,  in  the  wheel  of  our  example  to  §  88,  the  jet  or  layer  of  water  falling  on 
the  wheel  has  2£  feet  fall,  then  as: 

Q-  10.  and  e=  5,1,  rf1  =  °'1265  "  10  =  1^=0.157  feet. 

5,1.^5       8>°J$c 

If  now  we  allow  an  equal  thickness  for  the  exit  of  the  air,  then  the  least  distance  of  two 
buckets  becomes  0,314  feet,  or  3J  inches. 

Remark.  To  find  the  elbow  angle  J,  in  that  construction  of  bucket  which  is  based  on 
the  thickness  of  the  layer  of  water,  let  us  put : 

5  =  180°  —  CBJ1  =  180°  —  CBE,  —  ElBj}=  180°  —  ^90_  B-\  —  4,  =  90° -f  *-  —  <}>, 
Pt  N  d. 


but  sin.  <}>=:  • 


a 

2  a.sin.l 
2 


,  when  rf,  is  the  least  distance  between  two  buckets,  and 


o,  the  radius  of  the  division  circle. 
0,3 1 4 

feet,  hence  sin.  <f>  = ' —  = 

29  sin  j>*° 


For  th 

;°'314 
Mtffi 


last  example,  8  =  ' 
0,2482,  hence  *  = 


0,314,  a,  =  14,5 
22',  and  S  =  90° 


_|_  2°,  30'  —  14°  22' 


78 


§  91.  Sluices,  Pentroughs,  or  Penstocks. — The  method  of  bring- 
ing the  water  on  the  wheel  is  Q2 
of  no  small  importance.  Either 
the  water  falls  freely  out  of  the 
lead  or  trough,  or  it  is  pent  up 
by  a  sluice,  or  pentrough,  or 
penstock,  before  entering  the 
wheel.  In  the  former  case,  the 
velocity  of  entrance  depends 
on  the  inclination  of  the  trough 
or  the  height  of  fall.  In  the  second  case,  it  may  be  regulated  by 
adjusting  the  height  of  head  created,  and,  therefore,  this  latter 
method  should  be  preferred.  Fig.  192  shows  a  trough  without  a 
regulating  sluice ;  but  there  is  a  waste  board  at  F  by  which  the 
quantity  of  water  can  be  regulated.  If  the  water  flows  along  the 


178  SLUICES. 

trough  with  a  velocity  cr  and  if  the  fall  from  the  end  of  it  to  the 
centre  of  the  cell  =  A1?  the  velocity  _ 


if  Q  be  the  quantity  of  water,  and  F  the  sectional  area  of  the  water 
coming  on  the  wheel. 

The  penstock  (Fr.  vannes;  Ger.  SpannscTiutze]  is  either  vertical, 
horizontal,  or  inclined.     Fig.  193  shows  the  arrangement  of  a  hori- 

Fig.  193.  Fig.  194. 


zontal  sluice,  and  Fig.  194,  that  of  a  vertical  sluice.     The  construc- 
tion of  inclined  sluices  as  shown  in  Figs.  195  and  196.     The  one, 

Fig.  195.  Fig.  196. 


Fig.  195,  is  the  arrangement  general  in  the  Freiberg  district,  the 
sluice  being  raised  and  depressed  by  means  of  a  screw  S.  In  Fig. 
196,  a  simple  lever  is  used  for  these  purposes.  It  is  a  general  rule 
for  these  penstocks,  to  make  them  as  smooth  as  possible  inside,  and 
to  round  off  the  edges  of  the  orifice,  so  as  to  adapt  it  to  the  form  of 
the  contracted  vein,  that  the  resistance  may  be  the  least  possible.  If 
the  water,  after  passing  the  sluice,  fall  quite  freely,  and  if  we  can 
place  the  plane  of  the  orifice  at  right  angles  to  the  jet  of  water,  it 
becomes  then  advisable  to  make  the  orifice  as  in  a  thin  plate,  but 
in  that  case,  care  must  be  taken  that  partial  contraction  does  not 
occur,  for  this  gives  rise  to  an  obliquity  of  the  jet  (Vol.  I.  §  319). 

In  the  discharge  from  penstocks,  the  velocity  of  discharge  is  de- 
duced from  the  height  h1  by  the  formula  cl  =  $>  \/%g  h^  and  if  h2 
be  the  height  of  fall  after  passing  the  orifice,  to  the  centre  of  the 
cell,  then  the  velocity  of  entrance  c=  i/cf+Z  gh2=  \/Zg  (t2  ^1  + ^2)* 
If  we  take  the  velocity  co-efficient  <}»=  0,95,  then  <?=  * 


SLUICES.  179 

We  see  from  this,  that  for  equal  falls  the  velocity  of  entrance  must 
be  very  nearly  equal,  whether  it  flow  on  freely,  or  be  discharged 
from  a  sluice,  on  to  the  wheel. 

§  92.  That  the  water  may  enter  unimpeded  into  the  wheel  cells, 
it  must  not  come  in  contact  with  the  bucket  at  the  outer  circum- 
ference, but  nearer  to  the  inner  circumference  or  bottom  of  the  cells. 
Hence,  not  only  must  the  outer  edge  of  the  buckets  be  sharpened 
off,  but  the  layer  of  water  AC,  Fig.  197, 
must  be  so  directed  that  its  velocity  may  Fig  107. 

be  decomposed  into  two  others,  one  of 
which  is  in  the  direction  of  the  velocity  of 
the  wheel  Av  —  v,  and  the  other  in  the 
direction  AB  of  the  arm  or  wrist  of  the 
bucket.  As  we  may  assume  the  direction 
of  the  outer  element  of  the  bucket — the 
velocity  at  the  outer  circumference  of  the 
wheel  v,  at  right  angles  to  the  radius  AQ 
of  the  wheel — and  the  velocity  c  of  the 
water  coming  on  to  the  wheel,  to  be  given, 
we  shall  have  the  required  directiori%f 
the  water  layers  if  we  draw  through  v  a 
parallel  to  AS,  and  with  c  as  radius, 

describe  an  arc  from  A  as  centre,  and  draw  from  A  to  the  inter- 
section of  the  arc  with  the  parallel,  the  straight  line  Ac,  or  by  calcu- 
lation as  follows  : 

The  angle,which  the  velocity  v  of  the  circumference  makes  with 
the  outer  element  of  the  bucket  J1BJ=  v  JIB  =  t,  may  be  deduced 
from  the  elbow  angle  ABE  =  8,  and  the  division  angle  ACB  =  3, 
by  the  equation  t=*ACJB  +  BAC=  ^  +  90°  —  <j>,  and  hence  $=90° 

From  $,  v  and  c  we  have  the  angle  c  AB  =  4,  by  which  the  direc- 
tion of  the  layer  of  water  must  deviate  from  that  of  the  arm  of  the 
bucket,  in  order  that  the  water  may  enter  the  cells  unimpeded :  for 

sin.  4       v        ,    ,,       f 

=  _   and,  therefore, 

sin.  $       c 

sin,  4  =  v  8in'  *  =  *«»•(*— £j  (See  Vol.  I.  §  32.) 
c  c 

Again;  the  angle  c  AIT  of  the  direction  of  the  water  layer  to  the 
horizon,  is  vl  =  $  —  4  +  ®;  &  being  as  above  the  angle  ACS,  by 
which  the  point  of  entrancejof  theX4)water  on  the  wheel,  deviates 
from  the  summit  8. 

The  relative  velocity  Acl  =  c^  with  which  the  water  enters  the 

cells  is  Cl  =  *«**•(» -4). 
sin.  t 

Example.  Suppose  a  water  wheel,  the  velocity  at  the  circumference  of  which  t>=  10 
fret,  the  velocity  of  the  water  c=  10  feet,  the  elbow  angle  =  70$°,  the  division  angle 
£,  =  4$°,  and  the  point  of  entrance'of  the  water  deviates  12°  from  the  summit:  then 
*  =  90°  — (70§°  — 4i°)=24°,  and,  therefore,  sin.  4=|$  tin.  24°  =  0,271 16,  hence 
4  =  15°,  44'.  Thus,  that  the  water  may  enter  unimpeded,  the  deviation  of  the  layer 


180 


SuUiCES. 


must  be  15j°  from  that  of  the  arm  or  outer  element  of  the  bucket.     The  angle  of  incli- 
nation  to   the   horizon,  or  »,  =  24°  —  15j°  -f-  12°  =  20^°,  and  the   relative   velocity  is 

fi  =  l^L8_!_l^  =  5,303  feet. 

Remark.  It  is  generally  considered,  in  older  works  on  this  subject,  that  the  water  layer 
should  enter  the  wheel  in  the  duer.tion  of  the  arm  of  the  bucket;  but  this  rule  is  only  true  when 
v  =  0,  or  J  —  Bi  =  90°,  and  these  cases  never  occur.  The  deviation  ^  is  of  course  very 
small  for  a  wheel  revolving  slowly,  but  never  so  small  as  to  allow  of  our  assuming  it  as  0. 
When  the  water  enters  in  the  direction  of  the  outer  element  of  the  bucket,  the  bucket 
strikes  against  the  water,  and  throws  it  before  it  with  a  velocity  v  sin.  <p,  by  which  iris 
viva  is  lost,  and  water  spilt. 

§  93.  That  the  water  may  reach  the  wheel  with  the  direction 
required,  either! the  sluice-opening  is  laid  close  up  to  the  point  at 
which  the  water  is  to  enter  the  wheel,  and  the  sluice  is  set  at  right 

angles  to  the  direction  of  the 

Fiz.  1 98.  layers  of  water,  or^an  additional 

trough  is  laid  in  the  required 
direction  of  the  layer,  orTthe 
sluice  is  so  placed,  that  the 
direction  of  the  parabolic  curve, 
formed  by  the  water  in  its  free 
descent,  may  be  that  required. 
The  Fig.  198  shows  the  pen- 
trough  used -in  the  Freiberg  dis- 
trict, in  which  the  bottom  piece 
BD,  and  the  lower  part  of  the 
sluice-board,  are  set  obliquely  to 

the  direction  of  the  water  layer,  so  that  each  makes  an  angle  of 
about  14£°  with  the  direction  of  the  axis  of  it. 

In  order  to  find  the  direction  of  the 
sluice-board,  when  part  of  the  water 
falls  freely  into  the  wheel,  we  have  to 
recur  to  the  theory  of  projectiles, 
given  in  Vol.  I.  §  38,  &c.  From  the 
velocity  Ac  =•  c,  Fig.  199,  and  the 
angle  of  inclination  RAM  =  Vl  of  the 
required  direction  of  the  layer  to  the 
horizon,  the  vertical  co-ordinate  MO 
of  the  apex  of  the  parabola  is  : 

'  "'  . ,  and,    on    the    other 


Fisr  1 90 


hand,  the  horizontal  co-ordinate  : 


c2  sin.  2  * 


If,  now,  we  wish  to  place  the  sluice-aperture  at  any  point  P  of 
this  parabolic  curve,  and  if  we  know  the  height  MN  =  a,  of  this 
point  above  the  point  of  entrance  A,  then  for  the  co-ordinates  of 


this  point  OJV 
a   and 


and  NP 


we  have  the  formulas  :  x  =  x  — 


EFFECT  OF  IMPACT. 


and  for  the  angle  of  inclination  TPJ\T=  „,  which  the  parabola  makes 
with  the  horizon  on  this  point, 


PJY          PJY    -    y' 

The  plane  PK  of  the  sluice-board  must  be  set  at  right  angles  to 
the  tangent  PT;  and  thus  we  find  the  required  position  of  the°sluice 
board,  if  we  set  off  the  abscissa  OJVin  the  opposite  direction  OT, 
draw  PT,  and  erect  a  perpendicular  to  it  PK. 

If  the  sluice-aperture  be  set  at  the  apex' of  the  parabola,  then  the 
sluice-board  will  have  to  be  vertical. 

The  velocity  of  discharge  at  P  is  c0  =  v/c2  —  2  g  a,  and  the  cor- 
responding theoretical  pressure  height  h0  =  ~ a,  or  the  effective 

(-2  \ 

2 a),  when  the  orifice  is  nearly  rounded.     The 

breadth  of  the  sluice-orifice  is  made  a  little  less  than  the  breadth  of 
the  wheel. 

Example.  For  velocity  c  =  15  feet,  and  angle  »,  =  20i°  (see example  to  last  paragraph). 
the  co-ordinates  of  the  .parabola's  apex, are  3",  =0,0155.  15'  (fin.  20^°)"  =  0,43  feet, 
and  y,  =  0,0155  .  15*  .  *i«.  40$°  =2,33  feet.  If,  now,  the  centre  of  the  slmct-aperttire 
is  to  be  4  inches  =  0,333  feet  above  the  point  of  entrance,  then  the  co-ordinates  from  the 
centre  of  the  opening: 

I7TT 
0,/f.,  1=0,43  —  0,33  =  0,1,  y  =  2,33       _li-  =  1 ,!  1  feet,  and 

>|U,43 
0  2 
tang  »  = — : — —  9°,  58',  and  hence  the  inclination  of  the  sluice-board  to  the  horizon  is 

=  90°  _  r  =  90°  —  9°,  58'  =  80°,  2'. 

§  94.  Effect  of  Impact. — In  the  overshot  wheel,  the  water  acts 
in  some  degree  by  impact,  but  chiefly  by  its  weight.  We  determine 
the  effect  of  the  shock,  by  deducting  from  the  whole  effect  corre- 
sponding to  the  vis  viva  which  the  water  entering  the  wheel  possesses, 
the  mechanical  effect  retained  by  the  water  when  it  leaves  the  wheel, 
and  that  lost  by  the  oscillatory  and  eddying  motion  of  the  water  in 
the  cells.  The  velocity  of  the  water  leaving  the  wheel  may  be 
assumed  as  equal  to  the  velocity  t\  of  the  wheel  in  the  division  circle, 
and  hence  the  mechanical  effect  retained 

in  this  water  is  J-  Q  y.     The  mechanical 

effect  lost  by  the  oscillation  and  eddying 
motion  of  the  water  may  be  put  equal  to 

•+•  Q  y,  where  v2  is  the  velocity  suddenly 

lost  by  the  water  entering  the  wheel.  If, 
therefore,  et  be  the  velocity  Bc^  Fig. 
200,  of  the  water  entering  the  wheel,  the 
mechanical  effect  still  inherent  in  its  vis 


VOL.  II. — 16 


182  EFFECT  OF  IMPACT. 

But  the  velocity  cx  maybe  decomposed  into  two  others  Bv^  =  v^  and 
Bv2  =  v2,  of  which  i\  is  exactly  the  velocity  retained  by  the  water  as  it 
moves  on  with  the  wheel,  and,  therefore,  v2  is  the  velocity  lost.  If  we 
put  the  angle  cl  Bvv  which  the  direction  of  the  entrance  velocity  cl  of 
the  water  makes  with  a  tangent  B  i\  (the  direction  of  the  velocity 
of  the  circumference)  =  j»,  then  we  have  v22  =  c*  +  v*  —  2  c^  vl  cos. 
^,  and,  therefore,  the  mechanical  effect  in  question  : 

L   =  (C?  ~  ^  ~  **  ~  V*  +  2C1V*  C°8'  *  Qy  = 


9  9 

and  y  =  62,5,  L  =  2,008  (c,  cos.  /»  —  v,}  vt  Q  feet  Ibs.  - 
.  '.  It  is  evident  that  the  mechanical  effect  of  impact  is  so  much  the 
greater,  the  greater  ^  is,  and  the  less  ^  ;  and  by  comparing  with 
Vol.  I.  §  386,  it  follows  that  this  effect  is  a  maximum  when  vl  =  | 
cl  cos.  ft.     The  maximum  effect  corresponding  to  this  latter  ratio  is  £ 


;  or  when/«=0,  orco«./i  =  l,  then£=i  .-Q7.  As 

.  3  *  * 

^-  is  the  fall  due  to  the  velocity  <?„  it  follows,  that',  in  the  most  favor- 

*ff 

able  case,  the  effect  of  impact  is  only  half  the  available  effect.    Hence 

the  least  possible  part  of  the  fall  should  be  spent  to  produce  impact, 

as  much  as  possible  being  employed  as  weight.    Suppose,  for  instance, 

we  make  c.  cos.  p  =  vv  therefore,  c.  =  —  !-,  we  sacrifice  a  height  of 

cos.  /* 

fall  ~  -  ^  -  ,  without  having  any  mechanical  effect  in  return,  but 
2g  cos.  M8 

if  we  make  c,  =  -^--,  we  expend  four  times  that  fall,  viz  : 
cos.  /* 

-,  and  yet  we  have  only  : 
2 


and  lose  thereby  the  amount  of  fall  represented  by  :  tf  I 

(A  \    *.  2 

-  5  —  2  )  -±.,  and  even  if  we  assume  u  =  0,  or  cos.u=  l>the 
cos.  p*        /  2g 


2 


loss  of  fall  is  2  .  -L,  or  double  as  much  as  when  we  avoid  all  shock, 

or  bring  the  water  on  to  the  wheel  with  the  velocity  with  which  the 
wheel  revolves.  Again,  we  perceive^ hat  the  efficiency  of  the  wheel 
will  be  greater  the  less  v1  is,  or  the  slower  the  wheel  revolves.  It  is 
true  that  the  capacity  of  the  wheel,  its  width  e,  and,  therefore,  its 
height,  must  be  greater  as  the  velocity  of  revolution  v,  is  less ;  and 
as  the  journals  of  a  wheel  must  be  of  greater  diameter  the  heavier 
the  wheel  is,  and  as  the  moment  of  friction  increases  as  the  radius 
of  the  journal,  the  mechanical  effect  consumed  by  the  journal  friction 


EFFECT  OF  THE  WATER'S  WEIGHT.  183 

in  the  case  of  the  wheel  revolving  slowly,  may  be  greater  than  in 
one  moving  more  rapidly ;  and  hence  we  perceive  that  it  by  no  means 
follows  as  a  matter  of  course,  that  the  slower  a  wheel  revolves,  the 
greater  its  efficiency  will  be.  nn'cU-§  *?•/*•  i7'L- 

§  95.  Effect  of  the  Waters  Weight.— The  cells  of  a  water  wheel, 
when  filled,  form  an  annular  water  space  AB,  Fig.  201,  which  is 
termed  the  water- arc;  as  the  water 
enters  at  the  upper  part  of  this  arc,  Fig.  201. 

and  leaves  it  at  the  lower  end,  its 
height  h  is  the  effective  fall,  and,  there- 
fore, the  mechanical  effect  given  off 
by  the  weight  of  water  =  h  .  Q  y. 
The  height  of  the  water  arc  may  be 
subdivided  into  three  parts.  The  first 
part  HM  lies  above  the  centre  of  the 
wheel,  and  depends  on  the  angle  SCJi 
=  0,  by  which  the  point  of  entrance 
deviates  from  the  vertical  passing 
through  the  summit  of  the  wheel.  If, 
again,  we  put  the  radius  of  the  wheel 
CA  =  a,  the  height  of  the  upper  part 
of  the  water  arc  MH  =  a  cos.  ®.  The 
second  part  MK  lies  below  the  centre 
'of  the  wheel,  and  depends  upon  the 
point  D,  at  which  the  wheel  begins  to 
lose  water,  or  to  spill.  If  we  put  the 
angle  MCD  by  which  this  point  lies 
below  the  centre  of  the  wheel  =  x,  then 
this  second  height  MK  =  a  sin.  x.  The 

third  part  includes  the  arc  DB,  in  the  course  of  which  the  wheel 
empties  each  bucket  in  turn.  If  we  put  MOB,  the  angle  by  which 
the  point  B,  at  which  the  buckets  are  emptied,  deviates  from  M  the 
centre  of  the  wheel  =  xx,  then  the  height  KL  —  a  (sin.  x,  —  sin.  x). 
Whilst  now  in  the  two  upper  parts  of  the  arc,  the  water  has  its 
entire  effect,  it  communicates  only  a  part  of  its  mechanical  effect  to 
the  wheel  in  this  third  part,  because  here  it  gradually  quits  the 
wheel,  and,  therefore,  the  total  effect  of  the  water's  weight  must  be 
represented  by  a  (cos.  ®  +  sin.  x)  Q  y  +  a  (sin.  xx  —  sin.  x)  Ql  y,  when 
Ql  is  the  mean  quantity  of  water  effective  in  the  lower  division  of  the 
water  arc. 

If  we  combine  with  this,  the  effect  of  the  impact  of  the  water,  we 
have  the  total  mechanical  effect  of  an  overshot  water  wheel : 

_  ^_     -— -  •      ) 

-f-  a  (sin.  Xj — sin.*.)  Ql  y; 

or,  if  we  put  the  height  a  (cos.  ®  +  sin.  x)  of  the  part  of  water  arc -err  I 
taking  up  the  entire  effect  of  the  water  =  hv  and  the  remaining 

part  a  (sin.  x:  —  sin.  x)  =  A2,  and  the  ratio  —^  =  *,  then: 


184  EFFECT  OF  THE  WATER'S  WEIGHT. 


L=  Pv  =  WiW—Vi)'!    +  hl  +  x  h2)  Qy, 

\  9  ' 

and  the  force  at  circumference  of  the  wheel : 

— :p^'  +  *'  + -*')!"• 

Example.  The  velocity  of  entrance  of  the  water  on  an  overshot  wheel  of  30  feet 
diameter  is  c,  =  15  feet,  the  velocity  vt  of  the  division  circle  ^  9$  feet.  The  angle  by 
which  the  direction  of  the  water  layer  deviates  from  the  direction  of  motion  of  the  wheel 
at  the  point  of  entrance,  is  8^°,  and  the  deviation  of  this  point  from  the  summit  of  the 
wheel  is  12°.  The  deviation  of  the  point  where  the  wheel  begins  to  lose  water  from 
the  centre  of  the  wheel  X  =  58$°,  and  the  deviation  of  the  lowest  point  in  the  water 
arc  from  the  centre  of  the  wheel,  or  X,  =  70£°.  Lastly,  the  quantity  of  water  going  on 

to  the  wheel  Q  s=  5  cubic  feet  per  second,  and  K  =  _!  is  assumed  as  ^.     Required  the 

effect  of  the  wheel.  First,  the  effective  impact  fall—  0,031  (15  cos.  8£°  —  9$)  .  9$  = 
1,60  feet;  and  the  effective  weight  fall  is: 

15  (cos.  12°  +  sin.  58$°)  +  '/  (sin.  70^°  — sin.  58$°)=  15  (1,8307  -f  0,0450  =28,14 
feet,  and  hence  the  total  effect  of  the  wheel  is  (1,60  +  28,14)  .  5  .  62,5  =  9256  feet 

Ibs.  =  17  horse  power,  and  the  force  at  the  division  circle  is =  1000  pounds, 

nearly. 

§  96.  We  easily  perceive  from  this,  that  for  the  exact  determina- 
tion of  the  effect  of  the  weight  of  the  water  on  an  overshot  wheel, 
it  is  essential  to  know  the  two  limits  of  the  arc  in  which  the  wheel 

loses  its  water,  and  the  ratio  *  =  — J,  of  the  mean  quantity  of  water 

contained  in  the  cells  in  this  part  of  the  water  arc,  to  that  originally 
received  by  them,  On  this  subject  we  must  now  endeavor  to  ascer- 
tain the  necessary  rules. 

If  there  be  n  buckets  in  the  wheel,  and  if  it  make  u  revolutions 

per  minute,  there  are  presented  —  cells  per  minute  to  receive  the 
quantity  of  water  Q,  and,  therefore,  into  each  cell  there  goes  the 

quantity  V=  Q-T-  —  = -.      If  e  be,  as  hitherto,  the  width  of 

60         nu 
the   wheel,   then  the  section  of  the   prism    of  water   in    any  cell 

=  F  =Z=5°_9 

If  now  DEFG,  Fig.  201,  be  the  cell  at  which  the  water  begins  to 
spill,  then  the  section:  F0  =  segment  DEF  +  triangle  DFG,  or  'as 
the  triangle  DFG  =  triangle  DFJV* —  triangleJ^GJV,  then  F0  =  seg- 
ment DEF+  triangle  DFN—  triangle  DGJV.  If  we  put  the  area 
of  the  segment  DEF  =  S,  and  that  of  the  triangle  DFN=  D,  then 
the  triangle  DGJY*=  S+  D  —  F0;  but  as  the  triangle  DGJV  is  also 

,  .    D JV .  JVG        .    ,2  ,  ,  ,  .  •    .  . 

equal  to =  f  cr  tang.  %  nearly,  we  have  approximately 

(and  the  more  accurately  the  greater  the  number  of  buckets), 
tang,  x  = — — — -31 — °.  Thus  the  angle  MCD  =  >.,  corresponding 
to  the  point  at  which  the  wheel  begins  to  empty  itself,  is  determined. 


185 


EFFECT  OF  THE  WATER'S  WEIGHT. 


, 

Each  cell  will  have  emptied  itself  when  the  outer  elementAof  the 
bucket  becomes  horizontal;  if,  therefore,  the  angle  CBO,  which  this 
outer  element,  or  the  wrist  of  the  bucket,  makes  with  the  radius  =  x  , 
then  Xj  gives  us  the  angle  M  C#,  which  fixes  the  point  where  the 
cells  have  emptied  themselves.  In  order,  therefore,  to  find  the  effect 
of  the  water  on  the  discharging  arc7  let  us  divide  the  height  KL  =  a 
(sin.  >.j  —  sin.  x)  into  an  even  number  of  equal  parts,  indicate  the 
position  of  the  bucket  for  each  of  these  points  of  division,  draw  hori- 
zontal lines  through  the  sections  of  the  water  in  the  cells  for  each 
of  these  positions,  and  reckon  the  areas  Fv  Fv  F3  .  .  .  Fn  of  these 
sections.  The  mean  value  F  of  these  may  be  determined  by  the 
Simpsonian  rule,  putting 
F  =  F°+  Fn  +  4(F*  +  *3  +  •  •  •  +  FnJ  +  2(F2  +  F4  +  .  .  .  FnJ 

3n 

and  from  this  we  get  the  ratio  of  the  mean  quantity  of  water  in  a 
cell  in  the  discharging  arc  to  the  quantity  in  a  cell  before  it  begins 
to  empty  itself: 
=  Ql=  F  =  F0+ 
Q       F0 


Example.  There  are  300  cubic  feet  of  water  per  minute  supplied  to  a  water  wheel 
40  feet  in  diameter,  making  4  revolutions  per  minute.  What  is  the  effect  of  such  a 
wheel  ?  If  we  suppose  the  depth  of  the  shrouding  to  be  1  foot,  then  the  width  of  the 


wheel  = 


_ =_=  2,4  feet.     If  there   be    136   buckets  on  the  wheel,  the 

it  .  40.  1  .  4       4ir 


quantity  of  water  in  each  cell :  V  \ 


300 


75 
"136" 


the  section :  F 


0,5515 


square  feel 


4.  136 

144  .0,5515 


:  0,5515  cubic  feet,  and,  hence, 
33,09  square  inches.     By  accu- 


2,4  2,4 

rate  measurement  on  the  buckets  themselves,  as  they  are  represented  in  Fig.  2C2,  the 

Fig.  202. 


area  of  the  segment  J10BD  is  24,50  square  inches,  and  that  of  the  triangle  ^0FZ)=  102 
square  inches,  hence  lor  the  commencement  of  discharge : 

24,50  +102-30.09        93^  „  &       ^  x  =  ^  ^    ^ 

4  .  144  72 


tang.  X  = 


angle  at  which  the  wrist  of  the  bucket  meets  the  radius  is  X,  =  62°,  3(X,  and,  therefore, 
the  height  KA±  of  the  part  of  the  discharging  arc  retaining  water  is  =  a  (tin.  X,  —  tin.  x) 
=  20  (0,8870  —  0,7920)  =  1,79  feet.  If  within  this  height  we  delineate  three  relative 
positions  of  a  bucket,  we  find  by  measurement  and  calculations  the  section  of  the  water 

16* 


186 


EFFECT  OF  THE  WATER  S  WEIGHT. 


Fig.  203. 


space  in  the  bucket  for  these  positions  :  .F,  =  24,50  ;  F2  =  14,48,  and  F3  =  6,60  square 
inches.  As,  now,  the  section  at  the  commencement  is  F0  =  33,09,  and  at  the  end  it  is 
/"  =  0,  we  shall  find  the  ratio  : 

K_F  __  33,09  +  4  (24,50  +  6,60)  +  2  .  14,48  _  15,5375  _  0  469 

~fr0~~  1-2  .  33,09  ~    33,09 

If,  again,  the  water  enter  the  wheel  at  10°  below  the  summit,  and  with  a  velocity 

r,  =     Vl       so  that  the  water  acts  without  shock,  then  the  whole  mechanical  effect  given 

COS.  fj, 

off  by  the  wheel,  neglecting  the  friction  of  the  axle,  is  : 

L  —  a  [cos.  ©  +  sin.  X  +  0,469  (sin.  X,  —  sin.  X)]  X  5  X  62,25 

=  20  (0,9848+0,7920+0,469  .  0,085)6600=1.8167.6225=  11308  feet  lbs.  =  23,5 
horse  power. 

§  97.  Number  of  Buckets.  —  As  we   have   above  indicated,  the 
capacity  of  the  wheel  to  hold  water  should  be  made  as  great  as  pos- 

sible, or  the  buckets  should  retain 
the  water  as  long  as  possible,  so  that, 
cseteris  paribus,  the  maximum  effect 
of  the  fall  of  water  is  obtained  when 
the  buckets  are  placed  so  close,  that 
the  water  surface  AH,  Fig.  203,  in 
the  bucket  beginning  to  empty  itself, 
is  in  contact  with  the  bucket  ^1B1  D: 
next  above  it.  If  we  take  this  con- 
dition as  basis,  we  can  deduce  a  for- 
mula for  determining  the  number  of 
buckets.  From  the  angle  of  dis- 
charge MCF=  FAH  =  x,  and  the 
depth  of  the  shrouding  AF  =  d,  we 

have,  approximately,  FH  =  d  .  tang.  x.  If,  now,  we  assume  the 
division  circle  to  be  at  half  the  depth  of  the  shrouding,  we  may 
then  put  : 

2>1IT=  D,F  =  I  FH  =  \dtang.  x. 

If,  again,  we  introduce  the  angle  of  division  ECE^  =  /3,  and  the 
bucket  angle  A  CE  =  /3r  we  get  the  angle  A  CEl  =  ^l  —  j3,  and, 
approximately,  the  arc  D^  =  al  (j31  —  j3).  By  equating  the  two 
values  of  D^  we  have  «j  ($l  —  0)  =  J  d  tang,  x,  and,  therefore, 

0  =  0i  —  I      tang,  x, 


If  the  thickness  of  the  buckets  s 
of  buckets  would  be 

36Q° 


0,  then  of  course  the  number 


but  as  the  space  occupied  by  the  buckets  is  something  considerable, 
we  must  take  it  into  calculation,  and  thus  make  the  division  angle 

greater  by  an  amount  corresponding  to  an  arc  —  ,  or  we  must  take  : 


EFFECT  OF  THE  WATER'S  WEIGHT.  187 

If  we  introduce  tang,  x  =  S  +  f  ~  *Vand  F0  =  52-2,  then  we 

i  ^  n  w  e 

get: 


«!         <t,  d  nuald  e 

and,  therefore,  the  required  number  of  buckets  : 

2  *  M  at  d  e  —  60  Q 
~  [/3X  ax  d  +  s  d  —  (tf  +  D)]  w  e' 

If  we  put  d  e  =  —  -  —  —  -,  then  we  have  more  simply: 
2  a  u  « 


and  if  we  take  D  =  J  ft  «j  d,  then  : 


It  is  also  easy  to  perceive  that  the  angle  of  discharge  x  is  still 
more  increased,  when  (as  is  represented 
in  Fig.  204)  the  bucket  immediately  Fig.  204. 

following  that  which  is  about  to  dis-  " 
charge,  comes  in  contact  with  the  sur- 
face of  the  water  AHV  in  that  bucket, 
with  &  flat  surface  instead  of  a  corner  ; 
*>  or,  when  the  ivrist  of  the  bucket  is  not 
set  in  a  radial  direction,  but  in  such  a 
position,  that,  shortly  before  discharge 
commences,  it  is  horizontal.  In  this 
case,  the  segment  or  triangle  S=  Al 
Bl  Dj,  is  increased  by  a  triangle  Bl  Dl 

u       A  v,  %  +  D  —  F* 

H,  and  hence  tang,  x  =  -!—  -  -  -  ?, 

~2    & 

and,  therefore,  also  the  angle  of  discharge  a.  becomes  greater. 
Iron  buckets  are  always  rounded  at  the  corner  B. 

Example.  What  number  of  buckets  should  be  put  in  an  overshot  wheel  of  40  feet 
diameter,  and  I  foot  in  width,  giving  /S,°  =  4°  or  5.=  0,06981,  S  =  24,5  square  inches 
=  0,17014  square  feet,  and  the  thickness  of  the  buckets  *=  1  inch  =  0,0833  feet. 
According  to  our  formula  : 


*J    0.06981.19,54-0,0833  —  0,17014        0,.r)939 
which,  for  the  sake  of  facility  of  division,  we  may  take  152. 

Remark.  The  construction  of  bucket  shown  in  Fig.  204,  has  another  advantage,  viz.: 
that  for  it  a  less  width  of  wheel  is  necessary,  as  it  is  impossible  to  make  the  wheel  space 
=  four  times  the  capacity  of  the  water  space  (see  Vol.  II  §  92).  If  we  hf>re  introduce 
iS'=  |  a,  0  rf,  -}-  s  d1  tan8-  *>  an(l  as  S=  ^  cP  tang.  X  +  F0  —  D,  and  D  =  ^  a,  8t  d,  we 
obtain  :  tang.  X  =  fl|  '  _  9,  and  hence  : 


nua.de/  a,  «a,rfe 

If  we  neglect  the  thickness  of  the  buckets,  then  e=  |  .      60  ^  rf,ora  much  less  width 

of  wheel  than  was  assumed  at  §  92.  We  see  from  this  that  we  should  approximate  as 
nearly  as  possible  to  the  limits  of  bucket  construction,  of  which  we  have  now  been 
treating. 


188  EFFECT  OF  CENTRIFUGAL  FORCE. 

§  98.  Effect  of  Centrifugal  Force. — For  equal  velocity  of  the  cir- 
cumference, small  wheels  make  a  greater  number  of  revolutions  than 
large ;  but  the  uniform  motion  of  the  machine,  or  the  nature  of  the 
work  to  be  done,  as  sawing,  hammering,  grinding,  &c.  &c.,  require 
a  certain  velocity  of  the  wheel.     Hence  small  wheels  frequently 
make  a  great  number  of  revolutions  per  minute.     But  at  such  high 
velocities,  the  centrifugal  force  of  the  water  comes  into  play  to  such 
a  degree  that  the  inclination   of  the  surface  of  the  water  in  the 
buckets  to  the  horizon  is  considerable,  and, 
Fig.  205.  therefore,    the    discharge    commences    much 

earlier  than  would  be  the  case  if  the  wheel 
were  moving  slowly.  We  have  found  (Vol.  I. 
§  274)  that  the  surfaces  of  the  water  in  the 
buckets  are  a  series  of  cylindrical  hollows, 
the  common  axis  of  which  O,  Fig.  205,  runs 
parallel  to  the  axis  of  the  wheel,  and  lies  at  a 

i.  •  I,*     rn       7        <J  /30\2        2850 

height :   C0=  k  =  ?-  =  g  .  I—  )    =  — _ 

w2  \nu/  u2 

above  the  axis  C  of  the  wheel.  This  distance 
increases,  therefore,  inversely  as  the  square 
of  the  number  of  revolutions,  and  becomes 
small  for  a  great  number  of  revolutions. 
Hence  we  at  once  perceive  that  the  water-sur- 
face is  horizontal  only  at  the  summit  and^at , 
the  bottom  of  the  wheel,  and  that  at  a  givon 
point  M  above  the  centre  of  the  wheel  the  deviation  from  horizontal 
is  a  maximum.  . 

The  deviation  UAW=  AOC  =  x  for  any  point  A,  at  a  dictanoo 
ACM  =  x  from  the  centre  of  the  wheel,  is: 
AH          a  cos.  x 


For  a  point  Al  above  M,  x  is  negative,  and  hence  : 

tang.  %  = " If  we  lay  off  a  tangent  OA  from  0.  we  have 

k  —  a  sin.  x 

in  the  point  of  contact  A^  that  point  at  which  the  deviation  from 
horizontality  is  greatest,  or  where  x  is  a  maximum,  and  =  x,  sin.  x 

being,  however,  =  -. 

Exampk  1.  For  a  wheel,  making  5  revolutions  per  minute,  k  = =  114  feet,  the 

25 
radius  a  being  =  16  feet,  and  the  discharging  angle  X  =  50°.     Then : 

tang.  Y= 16  ""'  5U° —  =  10'285  ,  therefore,  y  =  4°,  39*,  so  that  the  surface  of  the 

114+I6««.SO       136,266 
water  deviates  in  this  case  4|  degrees  from  the  horizontal. 

Exampk  2.  For  a  wheel,  making  20  revolutions,  k  =  '^850  =   7,125.      If,    therefore, 

400 

«=  5,  X  =0°,  then  tang.  %  — ,  hence  £=  35°,  3'.     At  an  angle  of  44°,  34'  above 

/ ,  1  '2  5 
the  centre  of  the  wheel,  the  deviation  is  as  much  as  44°,  34'. 


EFFECT  OF  CENTRIFUGAL  FORCE. 


189 


But 


sin.  AOC 


sin.  x 


CO' 


Z.  sin.  x 


sin.  [90°  — | 
and  hence  follows : 

a  cos,  (x  +  x} 
k 

When  by  the  first  formula,  the  value 
of  x  +  x,  and  by  the  second,  the  value 
of  x  the  depression)  have  been  found, 
we  obtain  (by  subtraction  of  the  two 
angles)  x  =  (x  +  x)  —  x. 

At  the  end  A  of  the  angle  of  dis- 
charge, the  outer  end  of  the  bucket  co- 
incides with  the  water's  surface  A  Wv 
and,  therefore,  CA\Vl  —  ^l  +  xl^  this 
point  =  the  known)}ahgle  5,  depending 
on  the  form  of  bucket.  Hence 


CA*. 


#m.  Xl  = 


'  and  xx  =  6,—  Xl,  that 


F'g-  206. 


§  99.  If,  now,  we  take  into  consideration  the  effect  of  centri- 
fugal force,  as  is  obviously  necessary  in 
the  case  of  wheels  revolving  rapidly,  the 
formula  we  above  found  for  the  arc  of 
discharge  must  be  replaced  by  others. 
Let  A0  (Fig.  206)  be  the  point  at  which 
discharge  commences,-  MCA0  =  H0rfnG 
=  a.  the  angle  of  discharge;  H0J]0W0 
=  J)QOC  =  z  the  depression  of  the  wa- 
ter's surface  below  the  horizon,  or  the 
angle  CJ0W0  =  *.  +  x,  and  the  A  J0GJV0 
=  i  d  .  d  tang,  (x  +  *)  =  J  d2  tang. 
(x  +  'x).  If  we  now  put  the  contents 
of  the  segment  A^D0  —  S,  that  of  the 
A  AfoDy  =  D  ,  and  the  section  of  the 
body  of  water  =  F^  then  F0  -f  J  d2  tang. 
(x  +  z)  =  S+  D,  are,  therefore  : 


•*•»•  fSSgJcA*. 

'    =  ](.&**, 


is,  the  angle^by  which  the  end  of  the 
arc  of  discharge  deviates  from  the  centre 
of  the  wheel. 

If  the  height  H0H4  =  h4  =  a  (sin.  x, 
—  sin.  x),  Fig.  207,  of  the  arc  of  dis- 
charge, be  divided  into  4  or  6  equal 
parts,  and  the  filling  of  the  bucket  for 
corresponding  positions  be  determined,  we  can  again  find  the  ratio 


=  CUOW.C 
•txu 


190  EFFECT  OF  CENTRIFUGAL  FORCE. 

O         V 

x  =  _Ii  —  _  of  the  mean  contents  of  the  buckets  during  the  dis- 

charge, to  the  contents  before  discharge  commences,  and  so  cal- 
culate the  mechanical  effect  of  the  water  in  the  arc  of  discharge. 
For  this  the  above  formula  must  be  used  inversely.  In  this  case  x  is 
given,  hence  : 


tang.  x  =  __,  and  F=  S+  D  —  I  d*  tang,  (x  +  *). 

K  +  a  sin  x 

If  the  water  does  not  fill  the  entire  segment,  or  if  F  <  S,  or 
tang,  (x  -f  x)  >  A  then  we  must  put  : 

F=  segment  JIBD—  A  ADK, 
and  in  the  case  of  straight  buckets  : 
r       o      1^2     8* 

"  =  "°  —  2^  ai    '  \ 

sin.  (x  +  x) 

in  which  d  is  the  diagonal  AD  and  5j  is  the  angle  DAC  included 
between  this  and  the  radius  CA. 

Example.  A  small  wooden  water  wheel  (Fig.  207),  12  feet  high,  1  foot  depth  of 
shrouding,  and  4  feet  wide,  receives  1080  cubic  feet  of  water,  when  making  17  revolutions 
per  minute;  required  the  mechanical  effect  produced  by  it.  Here,  a  =  t>,  d  =  1,  e  =  4, 

a,  =  5,5,  Q  =  1^2  =  18,  UBS  17  ;  allowing  24  buckets, 

_  360°       150  and  p  108°      _  !£  —s  0,662  square  feet  If,  again,  D  =  0,652, 

~    24  24  .  17  .  4       68 


and  5=0,373,  then  tang,  (x  +  X)  =  °'373  +  °^  ~  =  0,363    X  2  =  0,726, 

OR  'SO 

therefore,  X  +  x  =  35°,  59^.     But  CO  =  k  =  ITl:  =  9,86  feet,  and  hence 

«  6  cog.  35°,  59'        0  4924    hence      _  29o   3(y  and  x  _  go  29/      ^ug  the  dis. 

9,86 

charge  commences  in  this  case  at  only  6^°  below  the  centre  of  the  wheel.  To  find  the 
point  at  which  the  discharge  is  complete,  we  have  in  the  present  case  (in  which  water 
hangs  in  the  bucket,  although  the  water's  surface  touches  the  outer  extremity  of  the 

bucket),  to  put  in  the  formula  sin.^  =  a  mg  instead  of  a,  the  radius  of  the  division 
circle  a,  =  5,5,  and  instead  of  fr,  the  angle  formed  by  the  outer  element  of  the  bucket  and 

the  radius,  and  which  is  here  79°,  14'.     Hence  :  sin.  y,  =  5'5  ""•  79°'  14/  ,   therefore. 

9,86 

X  =  5°.  59',  and  the  second  angle  of  discharge  X,  =  79°,  14'  —  5°,  W  =  73°,  15'. 
Hence,  the  height  of  the  arc  of  discharge,  A,=  a,  sin.  X,  —  a  sin.  X  =  5,5  sin.  73°,  15' 
—  6,0  sin.  6°,  2V  =  5,2666  —  0,6775  =  4,589  feet.  Dividing  this  height  into  4  equal 
parts,  \ve  determine  by  delineation,  by  measurement,  and  calculation,  the  corresponding 
three  intermediate  values  of  F.  The  results  arrived  at  are  -F,=  0,501,  F,  =  0,409, 
and  -F3  =  0,195,  and,  therefore,  the  required  ratio  of  the  sections  : 

x  _  f^  _  0,662+4  (0,50  1-f  0.1  95)  +  2  .  0,409  _n  53? 

^°  ~~  12.0,662 

and  the  mechanical  effect  produced  by  the  water  during  the  descent  of  the  arc  of  dis- 
charge :  Z,  =  x  .  hl  Qy  =  0,537  .  4,589  .  18  .  62,5  =  27,55  feet  Ibs.  If  the  water  fell 
with  a  velocity  of  20  feet,  20  degrees  under  the  summit  of  the  wheel,  in  such  a  direc- 
tion that  it  deviated  25°  from  the  tangent  at  the  point  of  entrance,  then  the  remaining 
effect  of  the  water's  pressure  : 

Z2  =(5,5  cos.  20°  +  6  sin.  6°,  29')  18  .  62,5=  5,854  .  1120  =6556  feet  Ibs. 

and  the  effect  of  impact,  the  velocity  in  the  division  circle  »,  being  -  L^-I  -  =  9,791 

feet  is  Z3=  0,031  (20  «w.25°  —  9.791)  9,791  .  18  .  62,5=2,611  X  1120  =  2974  feet 
Ibs.,  and  hence  the  whole  mechanical  effect  produced  is  : 

Z=iI-f  L2  +  L3=  12305  feet  Ibs. 


FRICTION  OF  THE  GUDGEONS.  191 

§  100.  Friction  of  the  Gudgeons.—  No  inconsiderable  portion  of 
the  mechanical  effect  of  overshot  wheels  is  lost  in  the  mechanical 
effect  absorbed  by  friction  on  the  gudgeons.  Let  the  weight  of  the 
water  wheel,  together  with  the  water  in  the  buckets,  =  G,  the  radius 
of  the  gudgeon  =  r,  then  the  friction  is  =/G,  and  the  velocity  at 

the  periphery  of  the  axis  =  v  =  ^Z  ,   and  hence  the  mechanical 

o(J 

effect  consumed  by  the  friction  of  the  gudgeons  =  /  G  v  =  *  u  rf  G 

=  0,1047  .  ufGr.  For  well  turned  gudgeons  on  good  bearings 
/=  0,075,  when  oil  or  tallow  is  well  supplied,  or/  =  0,054,  when  a 
constant  supply  of  best  oil  is  kept  up.  In  ordinary  circumstances 
of  the  application  of  a  black  lead  unguent,  /=  0,11.  The  weight  G 
of  the  wheel  must  be  determined  by  admeasurement  for  each  case. 
For  wheels  of  18  to  20  feet  in  height,  the  weight  has  been  found  to 
be  from  800  to  1000  times  the  number  of  effective  horse  power  in 
pounds.  The  wooden  wheels  of  Freiberg,  35  feet  in  height,  weigh, 
when  saturated  with  water,  nearly  44000  Ibs.  Being  20  horse  power, 
this  makes  upwards  of  2000  Ibs.  per  horse  power  for  the  weight  of 
the  wheel. 

The  effective  power  L  of  a  wheel  increases  as  the  weight  of  the 

wheel  increases,  as  the  proportion  of  the  bucket  filled  «  =     " 

dev 
increases,  and  as  the  number  of  revolutions  u  increases,  so  that,  in- 

versely, G  =  *  —  ,  in  which  i  is  a  co-efficient  to  be  ascertained  from 

*  u 

experience.      According  to  Rettenbacher,  a    small  iron  wheel,  the 

buckets  of  which  are  filled  £,  the  horse  power  being  6,3,  t  =  3432  Ib.s. 

For  the  Freiberg  wooden  wheels  of  20  horse  power,  *  =  2750 

Ibs.,  so  that,  for  a  first  approximation,  we  may  use  the  formula: 

G  =  3000  —  pounds. 

tU 

The  strength  of  the  gudgeons  depends  on  the  weight  of  the  wheel 
G,  and  thus  the  weight  has  a  twofold  influence  on  the  friction  (Vol. 
II.  §  87).  We  have  given  the  formula  : 

2  r  =  0,048  x/G  inches  =  0,00045  ^/G  feet,  for  the  strength_of 
gudgeons,  and,  therefore,  we  may  here  put  Gr  =  0,00142  \/G3, 
and  hence  the  mechanical  effect  consumed  by  the  friction  at  the 
gudgeons 

=  0,1047  uf.  0,00142  </&  =  0,00015 


Exampk.  What  amount  of  mechanical  effect  is  consumed  by  the  friction  of  a  wheel 
of  25000  Ibs.  weight,  with  gudgeons  of  6  inches  diameter,  the  wheel  making  6  revolu- 
tions per  minute.  Assuming  /=0,OS,  then/G  =0,08  X  25000  =2000  Ibs.,  and  the 
statical  moment  of  this=/Gr  =i  .  2000  =  500  feet  Ibs.,  and  the  mechanical  effect 
=  0,1047  .  6  ./Gr  =  314  feet  Ibs. 

Remark.  The  gudgeon  friction  of  a  water  wheel  may  be  increased  or  diminished 
according  to  the  manner  in  which  the  mechanism  for  transmitting  its  power  is  applied 
to  it.  If,  as  in  Fig.  20?,  the  power  P  and  the  resistance  Q  act  on  the  same  side  of  the 
wheel,  then  the  friction  on  the  gudgeons  is  diminished  by  an  amount  equal  to  the  resist- 


192  USEFUL  EFFECT  OF  WHEEL. 

ance  Q,  so  that  the  friction  is  less ;  but  if  the  power  and  resistance  work  on  opposite  sides 
(as  in  Fig.  209),  then  the  pressure  on  the  gudgeons  is  increased  by  an  amount  equal  to 

Fig.  208.  Fiz.  209. 


the  resistance  Q,  and,  therefore,  the  friction  is  as  much  greater  as  in  the  other  case  it  was 
less.  If  in  the  former  case  the  leverage  CB  of  the  resistance  be  made  equal  to  the 
leverage  of  the  power  Cd,  so  that  the  transmission  is  effected  by  a  toothed  wheel  on  the 
periphery  of  the  water  wheel,  as  shown  in  Fig.  171,  then  the  effect  of  the  power  on  the 
gudgeon  is  counterbalanced  by  the  resistance  (not  if  the  pinion  wheel  be  in  the  position 
shown  in  the  diagram,  but  when  this  is  placed  below  the  centre  of  the  wheel). 

§  101.   Total  E/ect.—The  total  effect  of  an  overshot  water  wheel 
may  be  put  : 


;  ;  \  Q7_fr 

)  a 


9 

or  when  the  water  enters  in  a  direction  nearly  tangential,  and  with 
a  velocity  equal  to  that  of  the  wheel  : 


Hence  for  a  given  weight  of  wheel,  the  total  effect,  as  well  as  the 
mere  water  power  effect,  is  a  maximum  when  v  =  0,  or  when  the 
wheel  revolves  with  the  least  possible  velocity.  This  condition  does 
not  hold  good  in  practice  :  for  the  dimensions  and  weight  of  water 
wheels  depend  on  the  power  they  give,  and  on  their  velocity,  and 
are  so  much  the  greater,  the  greater  the  effect  or  power,  and  the 
less  the  velocity  of  the  wheel. 

§  102.  Useful  Effect  of  Wheel—  Smeaton,  Nordwall,  Morin,  and 
others,  have  experimented  on  the  efficiency  of  overshot  water  wheels  : 
but  there  is  still  room  for  further  experimental  inquiry,  especially 
as  to  the  efficiency  of  well-constructed  large  wheels;  because  the 
effect  of  these  is  not  accurately  known,  and,  as  the  author  has  had 
occasion  to  verify,  their  efficiency  is  generally  under  estimated. 

Smeaton  made  experiments  with  a  model  wheel  of  75  inches  cir- 
cumference having  36  buckets,  and  found  for  20  revolutions  per 
minute  a  maximum  of  useful  effect  of  0,74.  D'  Aubuisson  mentions, 
in  his  work  on  hydraulics,  that  for  a  wheel  of  11J  metres  =  32  feet 
diameter,  with  a  velocity  of  8£  feet  per  second,  the  efficiency  was 
found  to  be  0,76.  Weisbach  has  tested  the  stamp-mill  wheels  at 


USEFUL  EFFECT  OF  WHEEL.  193 

Freiberg,  generally  7  metres  or  22'  —  9"  high,  and  3  feet  wide, 
having  48  buckets,  and  making  12  revolutions  per  minute,  and 
found  an  efficiency  of  0,78.  The  pumping  and  winding  wheels  of 
35  feet  diameter,  making  5  revolutions  per  minute,  give  an  efficiency 
of  0,80,  and  often  higher.  It  is  also  quite  ascertained  that  well- 
constructed  wheels  of  greater  diameter  than  the  above,  give  0,83 
efficiency,  the  losses  being  3  per  cent,  for  shock  at  entrance,  9  per 
cent,  by  too  early  discharge,  and  5  per  cent,  for  gudgeon  friction. 

Small  wheels  always  give  a  less  efficiency  than  larger,  not  only 
because  they  make  a  greater  number  of  revolutions,  but  because 
they  have  a  smaller  water  arc.  The  most  complete  experimental 
inquiry  on  water  wheels  is  that  of  M.  Morin,  "Experience*  sur  les 
Roues  hydrauliques  d  aubes  planes,  et  sur  les  Roues  hydrauliquex 
a  augets,  Metz,  1836."*  Of  these  experiments  we  can  here  only 
allude  to  those  made  on  three  small-sized  wheels.  The  first  of  these 
was  a  wooden  wheel  3,425  metres  =  10,6  feet  diameter,  with  30 
buckets,  and  giving  for  a  velocity  of  the  circumference  of  5  feet  per 
second,  an  efficiency  of  0,65,  and  the  co-efficient  v  =  0,775.  The 
second  wheel  was  only  2,28  metres  in  diameter  (7,47  feet)  —  of  wood, 
with  24  curved  plate  buckets.  With  a  velocity  of  5  feet  per  second, 
the  efficiency  of  this  wheel  was  =  0,69,  and  the  co-efficient  of  velo- 
city or  of  fall  =  v  =  0,762.  The  third  was  a  wooden  wheel  for  a 
»  tilt-hammer,  4  metres  diameter  (13  feet),  with  20  buckets,  and  1 
"metre  of  impact  fall  to  the  summit  of  the  wheel.  The  velocity  being 
5  feet  per  second,  the  efficiency  was  0,55  to  0,60,  and  the  velocity 
being  11J  feet  per  second,  its  efficiency  was  not  more  than  0,40, 
which  was  further  reduced  to  0,25,  when  the  velocity  rose  to  4  metres 
or  13  feet  per  second,  for  then  the  centrifugal  force  was  such  that 
the  water  could  not  properly  enter  the  buckets. 

Morin  deduces  from  his  experiments  that  for  wheels  of  less  than 
6'  —  6"  diameter,  having  a  maximum  velocity  of  6  feet  per  second 
at  the  periphery,  and  for  wheels  of  more  than  6'  —  6"  diameter; 
having  a  maximum  velocity  of  8  feet  per  second,  the  co-efficient  » 
of  the  pressure  fall  averages  0,78,  and,  therefore,  the  useful  effect 
of  these  overshot  wheels,  exclusive  of  friction  on  the  gudgeons,  is  : 


Pv 


0,78  A)  Q  y, 


h  being  the  height  of  the  point  of  entrance  above  the  foot  of  the 
wheel,  or  0,78  h  represents  the  mean  height  of  the  arc  holding  water. 
This  co-efficient  v  =  0,78  is,  however,  only  to  be  used  when  the  co- 
efficient (  representing  the  extent  of  filling  of  the  buckets  is  under  £  ; 
it  must  be  made  1,65  when  *  amounts  to  nearly  §.  In  the  case  of 
wheels  of  great  diameter  i  is  certainly  higher.  For  the  Freiberg 
wheels,  for  example,  it  is  0,9.  Morin  further  deduces,  that  when 
wheels  have  a  greater  velocity  of  revolution  than  6'  —  6"  per  second, 
or  if  the  buckets  be  more  than  f  filled,  a  definite  co-efficient  *  for 

[*  The  author  was  apparently  unacquainted  with  the  experiments  made  by  a  com- 
mittee of  the  Franklin  Institute.  —  AM.  ED.] 
VOL.  II.—  17 


194  AMERICAN  EXPERIMENTS. 

the  water-arc  cannot  be  given,  because,  then,  small  variations  or 
deviations  in  v  and  *  have  considerable  influence  on  the  amount  of 
the  useful  effect.  It  must,  however,  be  remarked,  that  it  is  not  the 
velocity,  but  the  number  of  revolutions  u  (Vol.  II.  §  98),  which  de- 
termines this  limit:  for  high  wheels  with  6'  —  6"  velocity  at  circum- 
ference, give  a  great  and  tolerably  well-ascertained  useful  effect. 

[American  ^Experiments. — The  most  important  of  the  deductions 
from  the  experiments  on  water  wheels,  made  in  1829-30,  by  the 
Committee  of  the  Franklin  Institute,  using  a  wheel  20  feet  in  diame- 
ter, and  with  a  head  and  fall  varying  from  20J  to  23  feet,  may  be 
stated  as  follows: — 

1.  "In  running  a  large  overshot  wheel  to  the  best  advantage,  84 
per  cent,  of  the  power  may  be  calculated  upon  for  the  effect." 

2.  "  The  velocity  of  the  overshot  wheel  bears  a  constant  ratio  for 
maximum  effects  to  that  of  the  water  entering  the  buckets,  this  rajtio 
being  at  a  mean  about  ,55  or  ^ths."* 

3.  "  Without  diminishing  the  ratio  of  effect  to  power  more  than 
2  per  cent.,  we  may  so  arrange  a  high  overshot  wheel  as  to  increase 
the  velocity  of  its  periphery  from  4J  to  Q^th,  and  probably  even  to 
7£  feet  per  second." 'f    * 

As  the  quantity  of  work  done  by  a  given  wheel,  when  the  ratio  of 
effect  to  power  is  the  same,  must  evidently  depend  on  the  velocity 
of  the  wheel,  it  must  be  advantageous  to  run  the  wheel  with  the 
highest  velocity  within  which  that  ratio  can  be  kept  constant,  or  nearly 
so,  that  is,  from  6  to  7  feet  per  second. 

The  ratio  of  effect  to  power  with  "centre  buckets"  was  found  to 
be  .78  of  that  with  "elbow  buckets,"  owing  to  the  water  sooner 
leaving  the  former  than  the  latter. 

When  air-vents  are  used  they  involve  a  loss  of  effect  with  centre 
buckets,  but  scarcely  vary  the  action  of  elbow  buckets.! 

4.  With  a  wheel  15  feet  in  diameter  "84  per  cent,  of  the  power 
expended  may,  as  before,  be  relied  on  for  the  effect,"  but  when  the 
heads  bore  to  the  falls,  or  heights  of  wheel,  a  proportion  as  great  as 
1  to  5  or  1  to  4,  the  ratio  of  effect  to  power  was  reduced  as  low  as 
,80  and  even  ,75. 

An  overshot  wheel  of  10  feet  in  diameter  gave  with  heads  of 
water  above  the  gate  varying  from  ,25  to  3,75  feet,  a  ratio  decreas- 
ing from  ,82  to  ,67  of  the  power;  and  with  a  wheel  6  feet  in  diame- 
ter, the  ratios,  under  like  variations  of  the  head  of  water  above  the 
gate,  varied  from  ,83  to  ,64.  The  same  general  law  of  a  decrease 
of  ratio  of  effect  to  power,  according  as  the  proportion  of  head  to  the 

•  [Mr.  Elwood  Morris  (see  Journal  Franklin  Institute  Vol.  IV.,  3d  series,  p.  222)  ascer- 
tained, by  direct  experiment  on  three  excellent  overshot  flouring  mill  wheels,  with  all 
the  modern  improvements,  that,  calculating  by  the  whole  head  and  fall,  while  they  ran 
at  their  usual  pace,  and  with  everything  in  order,  they  required  "788  cubic  feet  of  water 
falling  one  foot  per  minute,  to  grind  and  dress  one  bushel  of  wheat  in  an  hour."  This  is  an 
expenditure  of  power  =s  49,250  feet  Ibs.  per  minute  =  1^  horse  power. — AM.  ED  ] 

t  Journ.  Frank.  Inst.,  3d  series,  Vol.  I.,  pp.  149  and  154. 

t  Ibid,  p. 221.  §  Ibid, p.  223. 


HIGH-BREAST  WHEEL.  195 

total  head  and  fall  increases,  may  be  traced  in  the  action  of  differ- 
ent wheels. 

Thus  the  wheel  having  a  diameter  of 
15  feet,  and   mean  proportion  of  head  to  head  and  fall  =  ,063 

gave  ratio  of  effect  to  power  =  ,841 
20  feet,  and  mean  proportion  of  head  to  head  and  fall  =  ,067 

gave  ratio  of  effect  to  power  =  ,838 
6  feet,  and  mean  proportion  of  head  to  head  and  fall  =  ,072 

gave  ratio  of  effect  to  power  =  ,801 
10  feet,  and  mean  proportion  of  head  to  head  and  fall  =  ,079 

gave  ratio  of  effect  to  power  =  ,795. 

A  still  further  increase  of  proportion  until  the  head  was  45  per 
cent,  of  the  head  and  fall  gave,  in  the  case  of  the  6  feet  wheel,  a 
ratio  of  effect  to  power  only  ,604.  Indeed,  it  is  easy  to  understand 
that  all  that  head  of  water  above  the  bottom  of  the  bucket  which 
exceeds  what  is  necessary  to  give  the  water  the  same  velocity  as 
that  of  the  wheel,  can  act  only  by  creating  impact,  and,  therefore, 
must  be  considered  so  much  of  a  head  destined  to  produce,  not  pres- 
sure, but  percussion,  and  the  co-efficient  of  effect  of  any  water  de- 
livered under  such  increased  head,  must  be  undershot  co-efficient, 
which  experiment  proved  to  be  ,285.] 

§  103.  High-Breast  Wheel — The  overshot  wheel  frequently  re- 
ceives the  water  lower  than  the  point  we  have  above  indicated,  at  a 
>  point  somewhat  nearer  the  centre  of  the  wheel.  These  are  called 
by  the  French  roues  par  derridre,  by  the  Germans  ruckenschldgige 
Radar.  The  lead  or  water-course,  or  the  pentrough,  passes  above 
the  wheel  in  the  case  we  have  discussed.  For  high-breast  wheels 
the  pentrough  is  below  the  summit  of  the  wheel,  or  the  diameter  of 
the  wheel  is  greater  than  the  total  water-fall.  In  the  overshot 
wheel,  the  wheel  revolves  in  the  direction  in  which  the  water  is  led 
on  to  it.  In  the  high-breast  wheel,  it  revolves  in  the  opposite  direc- 
tion. High-breast  wheels  are  erected  more  especially  when  the 
level  of  the  water  in  the  tail-race  and  pentrough  are  much  subject 
to  variation;  because  the  wheel  revolves  in  the  direction  in  which 
the  water  flows  from  the  wheel,  and,  therefore,  backwater  is  less 
disadvantageous,  and  because  penstocks  or  sluices  can  be  applied 
that  admit  of  an  adjustable  point  of  entrance  of  the  water  on  the 
wheel,  or  of  maintaining  a  given  height  between  the  water  in  the 
pentrough  and  in  the  tail-race ;  and  even  for  different  conditions  of 
the  water-course,  the  same  velocity  of  discharge  and  of  entrance  of 
the  water  can  be  maintained.  Penstocks,  or  sluices  for  these  wheels 
are  represented  in  Figs.  210  and  211,  in  which  the  shuttle  JIB  is 
made  to  slide  or  fold  as  an  apron,  to  open  more  or  fewer  apertures 
as  required.  In  Fig.  210  JIB  is  made  concentric  with  the  circum- 
ference of  the  wheel,  in  order  that  the  aperture  Jl  may  direct  the 
water  into  the  wheel  for  all  positions  of  the  apron.  The  motion  of 
the  sluice  or  apron  is  effected  by  the  pinion  and  rack  AD  and  C. 
In  Fig.  211,  the  water  flows  over  the  top  Jl  of  the  sluice-board, 
which  is  adjusted  in  a  manner  similar  to  that  above  described.  In 


196 


HIGH-BREAST  WHEEL. 


order,  however,  that  the  water  may  come  on  to  the  wheel  in  a  pro- 
per direction,  a  set  of  stationary  ^rwcfe-buckets  EF,  is  laid  between 


Fig.  210. 


Fig.  211. 


Fig.  212. 


the  wheel  and  the  sluice-board,  and  this  latter  slides  over  them. 
The  guide-buckets  must  have  a  certain  position,  that  the  water  may 

not  strike  on  the  outer  end  of  the 
wheel-buckets.  If  Ac,,  Fig.  212, 
be  the  direction  of  the  outer  element 
of  the  wheel-bucket,  and  if  Av,  re- 
present in  magnitude  and  direction 
the  velocity  of  this  element  A,  then, 
exactly  as  in  Vol.  II.  §  92,  the  re- 
quired direction  Ac  of  the  water 
entering  the  wheel  is  obtained  by 
drawing  vc  parallel  to  Aev  and 
making  Ac  equal  to  c,  the  velocity 
of  the  water  entering,  as  deduced 
from  the  height  of  the  water's  sur- 
face above  A.  If  h  be  the  depth 
of  A  below  the  surface  of  the  pen- 
trough,  then  c  =  0,82  v/2  gh  at 
least,  as  in  the  discharge  through 
short  additional  tubes  (Vol.  I.  §  323),  and  when  the  guide-buckets 
are  rounded  off  on  the  upper  side,  then  c  =  0,90  */2gh.  If  the 
guide-curves  be  made  straight,  then  they  are  to  be  laid  in  the  direc- 
tion CAD,  but  if  curved  buckets  AE  are  adopted,  which  has  the 
advantage  of  gradually  changing  the  direction  of  the  water's  flow, 
then  they  are  made  tangents  to  AD  at  A,  by  raising  AO  perpen- 
dicular to  AD,  and  describing  an  arc  AE  from  0  as  a  centre. 

As  for  different  depths,  the  pressure  (h)  is  different,  and  the  velo- 
city due  (c)  is  also  different,  the  construction  must  be  gone  through 
separately  for  each  guide-bucket.  The  velocity  of  entrance  is  usually 
1*  to  10  feet,  and  the  velocity  of  the  wheel  is  \  c  to  f  c  at  the  most. 
The  construction  is  to  be  gone  through  for  the  mean  level  of  the 


BREAST  WHEELS. 


197 


water  in  the  trough,  that  the  deviation  in  cases  of  high  and  low 
water  may  not  be  excessive. 

From  this  kind  of  sluice  the  air  escapes  less  readily  than  in 
others,  and,  therefore,  the  sluice  is  made  narrower  than  the  wheel, 
or  the  wheel-buckets  are  specially  ventilated,  that  is,  the  floor  of  the 
wheel  is  perforated  with  air-holes.  It  is  not  advisable  to  make  the 
buckets  too  close,  but  rather  to  keep  the  water  in  the  wheel  by  an 
apron,  than  by  making  the  angle  of  discharge  too  great,  for  in  this 
latter  case  the  guide-buckets  extend  over  too  large  an  arc  of  the 
wheel,  or  become  long  and  contracted,  and  so  occasion  loss  of  fall. 

As  to  the  efficiency  of  high-breast  wheels,  it  is  at  least  equal  to 
that  of  the  ordinary  overshot.  From  the  advantageous  manner  in 
which  the  water  is  laid  on  to  them,  it  is  not  unfrequently  greater 
than  in  overshot  wheels  having  the  same  general  proportions.  For 
a  wheel  30  feet  high,  having  96  buckets,  the  entrance  of  the  water 
being  at  a  point  50°  from  the  summit,  the  velocity  at  circumference 
being  5  feet  per  second,  and  that  of  the  water's  entrance  8  feet, 
Morin  found  the  efficiency  ^  =  0,69,  and  the  height  of  the  arc 
holding  water  =  0,78  .  ^.* 

§  104.  Breast  Wheels.  —  These  wheels  are  either  ordinary  bucket 
wheels,  or  they  are  wheels  with  paddles  or  floats  confined  by  a  stone 
curb  or  wooden  mantle  (Vol.  II.  §  83).  As  by  a  too  early  discharge 
of  the  water  from  the  buckets,  the  greatest  loss  of  fall  or  of  mechani- 
cal effect  takes  place  in  the  lower  half 
of  the  wheel,  it  is  evident,  that  cseteres  Fis-  213- 

paribus,  breast  wheels  must  have  a 
smalkr  efficiency  than  overshot  or  high 
breast  wheels.  Upon  these  grounds  the 
fall  must  be  carefully  preserved  for 
breast  wheels,  or  the  water  kept  on  the 
wheel  to  the  lowest  possible  point. 
Hence  the  angle  of  discharge  for  their 
buckets  is  made  great,  or  the  water  is 
even  introduced  from  the  inside  of  the 
wheel,  as  shown  in  Fig.  213,  or,  as  is 
the  better  plan,  the  wheel  is  enveloped 
by  a  mantle  or  curb,  and  the  buckets 
or  paddles  are  set  more  or  less  radially. 
The  curb  must  not  be  at  more  than  from  £  an  inch  to  an  inch  from 


*  [With  a  high  breast  wheel  20  feet  in  diameter,  water  let  on  1  7  feet  above  the  bottom 
of  the  wheel,  under  a  head  of  9  inches,  the  Committee  of  the  Franklin  Institute  found 
that  elbow  buckets  gave  a  ratio  of  effect  to  power  of  ,731  at  a  maximum,  and  centre 
buckets  ,653.  Admitting  the  water  on  at  a  height  of  13  feet  8  inches,  the  elbow  buckets 
gave  ,658  and  the  centre  buckets  ,628.  At  10,96  feet  above  the  bottom  of  the  wheel,  the 
water  produced  on  elbow  buckets  with  a  head  of  4,29  feet,  a  ratio  of  ,544,  and  centre 
buckets  ,329,  with  the  gate  7  feet  above  the  bottom  of  the  wheel,  and  a  head  of  2 
of  water,  this  "  low-breast"  wheel  gave  a  ratio  of  ,62  for  elbow  buckets  and  ,531  for  centre 
buckets.  At  a  height  of  3  feet  8  inches  above  the  bottom  of  the  wheel,  and  one  foot  head 
above  the  bottom  of  the  gate,  elbow  buckets  gave  ,555,  centre  buckets  ,533.—  AM.  tn.J 

17* 


r- 
t 


198  OVERPAID  SLUICES. 

the  wheel,  that  as  little  water  as  possible  may  escape  through  the 
intermediate  space.  As  the  buckets  or  floats  of  wheels  inclosed  by 
a  curb  are  not  intended  for  holding  water,  they  may  be  placed 
radially,  but,  in  order  that  they  may  not  throw  up  water  from  the 
tail-race,  it  is  advisable  to  give  the  outer  element  of  the  float  or 
bucket  such  a  position  that  it  may  leave  the  water  vertically.  As 
to  the  number  of  floats,  it  is  advantageous  to  have  them  numerous, 
not  only  because  by  this  means  the  loss  of  water  by  the  play  left 
between  the  wheel  and  the  mantle  or  curb  is  smaller,  but,  also, 
because,  by  putting  them  close  together,  the  impact-fall  is  rendered 
less,  and  the  pressure-fall  increased.  The  distance  between  two 
floats  is  generally  made  equal  to  the  depth  of  the  shrouding  d,  or  it 
is  taken  at  from  10  to  15  inches,  or  one  of  the  rules  above  given 
(Vol.  II.  §  88)  may  be  applied  for  fixing  the  number  of  buckets. 
It  is,  however,  essential  that  breast  wheels  be  well  ventilated,  so 
that  the  air  can  escape  outwards,  because  in  them  the  stream  of 
water  laid  on  is  nearly  as  deep  as  the  whole  distance  between  two 
buckets.  Air-holes  or  slits  mustr  therefore,  be  left  in  the  floor  to 
prevent  the  air  from  interfering  with  the  water's  entrance.  This  is 
so  much  the  more  necessary  in  these  wheels,  as  they  are  usually 
arranged  to  be  filled  to  J  at  least,  even  f  of  their  capacity. 
Breast  wheels  are  generally  adopted  for  falls  varying  from  5  to  15 
feet,  and  for  supplies  of  from  5  to  80  cubic  feet  per  second. 

Remark.  Theoretical  and  experimental  researches  on  the  subject  of  breast  and  under- 
shot wheels,  with  the  water  laid  on  from  the  inside,  have  been  made  in  Sweden,  and 
are  given  in  detail  in  a  work  entitled  "  Hydrauliska  Forsok,  etc.,  of  Lagerhjelm,  of  For- 
selles  och  Kallstenius,  Andra  Uelen,  Stockholm,  1822."  Egen  describes  a  wheel  of  this 
kind  in  his  "  Untersuchungen  uber  den  Effect  einiger  Wassericerke,  &c ,  Berlin,  1831.'f  The 
efficiency  of  the  wheel  was  not  more  than  59  per  cent.,  although  the  fall  was  13,42  feet. 
A  wheel  on  the  same  model  was  erected  in  France,  but  only  t>' —  6"  in  diameter.  M. 
Mallet  experimented  on  this  wheel  (see  "Bulletin  de  la  Societe  d' Encouragement,  No. 
•.282)."  and  found  its  efficiency  60  per  cent.  It  would  appear,  therefore,  as  Egen  justly 
observes,  that  these  wheels  are  seldom  to  be  adopted.  They  can  have  only  a  limited 
width,  and  cannot  be  so  substantial  as  those  receiving  the  water  on  the  outside. 

§  105.  Overfall  Sluices. — The  mode  of  laying  on  the  water  to 
breast  wheels  is  very  various.  The  overfall  sluice,  the  guide-bucket 
sluice,  and  the  ordinary  penstock  are  in  use.  The  water  is  seldom 
allowed  to  go  on  quite  undirected.  In  the  overfall  sluices,  shown  in 
Figs.  214  and  '215,  the  water  flows  over  the  saddle-beam,  or  lip  A 
of  the  sluice ;  but  that  it  may  flow  in  the  required  direction,  the  lip 
is  rounded,  or  a  rounded  guide-bucket  AB,  Fig.  215,  is  appended 
to  it.  This  guide-bucket  AH,  Fig.  216,  is  curved  in  the  form  of  the 
parabola,  in  which  the  elements  of  water  lying  deepest  move ;  for  if 
it  were  more  curved,  the  water-layer  would  not  lie  to  it,  and  if  it 
were  less  curved,  the  width  of  the  guide,  and,  therefore,  the  friction, 
would  be  unnecessarily  increased,  and  the  water  would  reach  the 
wheel  less  in  the  direction  of  a  tangent.  According  to  the  theory 
of  discharge  over  weirs  (Vol.  I.  §  317),  if  e1  be  the  breadth  of  the 
weir,  Aj  the  height  above  the  sill  or  lip  If  A,  Fig.  216,  and  /*  the  co- 
efficient of  discharge,  then  Q=  ^  ue1h1  v/2#Aj ;  but  if  the  quantity 


OVERFALL  SLUICES. 
Fig.  214. 


199 


laid  on  Q,  and  breadth  of  the  aperture  ^  (which  is  only  3  or  4 

Fig.  215.  Fig.  216. 


inches  less  than  the  width  of  the  wheel  e)  be  given,  then  the  head 
for  the  discharge  : 


.  . 

Again,  the  velocity  c  of  the  water  entering  the  wheel  at  B  is  deter- 

mined by  its  proportion  %  =  -  to  the  velocity  of  the  wheel,  and, 

v 
hence,  the  fall  necessary  for  communicating  this  velocity  : 


. 

on  account  of  absorption  of  fall  by  discharge,  h2  —  1,1  •  —x  —  :  *1S 


200  PENSTOCK  SLUICES. 

generally  made  =  2,  and,  therefore,  Ji2  =  4,4  —  .     From  Til  and  Jt2 

we  deduce  the  height  AM  of  the  lip  of  the  guide-curve,  k=  h2  —  hv 
and  if  the  total  fall  HD  =  A,  there  remains  for  the  head  available 
as  weight  on  the  wheel  MD  =  EF  =h3=h  —  Ji2.  Again,  we  have 
from  the  theory  of  projectiles,  the  angle  of  inclination  TBM  of  the 
guide-curve's  end  to  the  horizon  determined  by  the  formula 


.,       h      f          in.a 
2g 
and  the  length  or  projection  MS  of  the  guide-curve  is  : 


t 

Lastly,  if  the  very  desirable  condition  of  bringing  the  water  tan- 
gentialty  on  to  the  wheel  is  to  be  fulfilled,  the  radius  of  the  wheel 
CB=  CF=  a  is  determined  by  the  equation  : 

a  (1  —  cos.  a  =  h  —  A2,  therefore,  a  =  -  ?-. 

1  —  cos.  a 

Inversely,  the  central  angle  BCF  =  o2  of  the  water  arc  is  determined 

by  cos.  aj  =  1  --  -,  and  when  the  latter  condition  is  not  ful- 
a 

filled,  or  oj  is  not  made  =  a,  then  the  deviation  of  the  direction  of 
the  water  entering  the  wheel  from  the  direction  of  the  motion  of  the 
bucket  on  which  it  impinges  :  5  =  al  —  0. 

Example.  Suppose  for  a  breast  wheel  the  quantity  of  water  laid  on  by  an  overfall 
sluice,  Q=  6  cubic  feet,  the  total  fall  h—  8  feet,  and  the  velocity  of  the  periphery  =  5 
feet,  also  the  ratio  of  the  buckets  filling^?,  then  for  a  depth  of  wheel  =  1  foot,  the 

requisite  width  on  the  breast  e  =  |  .  _  =  _  !  _  —  3  feet.     And  if  the  breadth  of  the 

(In        y  .  1  .  T> 
aperture  be  made  2|  feet,  and  ft  =  0,6,  then  the  height  at  which  the  water  stands 

^  =  0,3093   (      6  |()    =  0*76  feet.     If  we  take  x  =  f,  then  the  fall  necessary  to 

generate  the  velocity  c  with  which  the  water  enters  the  wheel  : 

c  =  |  .  5  =  8  feet,  /ia=  1,1  .  0,0155  .  8"=  1,1  feet  nearly,  and,  therefore,  the  height  of 

the  lip  of  the  guide  k=  1,1  —0,76  =0,34  feet  =4.  08  inches.     Again,  for  the  angle  of 

inclination  of  the  end  of  the  guide-curve  :  tin.  a  =    i—  _  =  0.5539,  and  hence  a  =  33° 

W    1,1 

38',  and  the  breadth  of  the  lip  of  the  guide-curve  /=  1,1  tin.  67°,  16'  =  ]  foot  nearly. 
That  the  water  may  enter  the  wheel  tangentially,  the  radius  of  the  wheel  would  have 
to  be 

a=    ft_;,2  g-1,1        =41)06 

1  —  cos.  a        1  —  cos.  33°,  38' 

but  if  we  limit  the  size  of  the  wheel  to  25  feet  diameter,  or  make  a=  12.5,  then  the 
central  angle  a,  of  the  water-arc  is  cot.  a,  =  1  --  '  —  0,45,  or  a,  =  63°,  16',  and 

the  deviation  of  the  direction  of  motion  of  the  water  from  that  of  the  wheel  at  the  point 
of  entrance  :  a,  —  «  =  63°,  16'  —  33°,  38'  =  29°,  38'. 

§  106.  Penstock  Sluices.  —  Fig.  217  shows  the  manner  of  laying 
on  the  water  on  a  breast  wheel  by  the  ordinary  penstock.  The 
sluice-board,  which  is  placed  as  close  to  the  wheel  as  possible,  is 
made  of  such  thickness  and  form  at  the  lower  edge,  as  obviates 


PENSTOCK  SLUICES.  201 

contraction.     For  the  same  reason  the  end  AB  of  the  bottom  of  the 

Fig.  217. 


pentrough   or  lead,  is  formed  with   a  parabolic  lip.     The   height 
BE=  DF=  h2,  Fig.  218,  of  the  end  of 
^the  curve  depends  on  the  total  fall  h,  and  Fig-  2 is. 

>on  the  velocity  height 


and  is  determined  by  the  formula 
7?2  =  h  —  Aj  :  hence,  as 

cos.  a  =  -—  =  a         2  ,  ^he  corresponding 
C-D  a 

central  angle  BCF  =  o  =  1  _  ^~ 


If,  therefore,  the  water  is 

to  be  taken  on  tangentially,  the  inclination  TBM  of  the  layer  of 
water  to  the  horizon  is  to  be  put  =  a,  and,  hence,  we  determine  the 
co-ordinates  MA  =  k  and  BM=  I  of  the  apex  of  the  parabola  A  by 
c2  sin.  o2  T  -,  c2  sin.  2  a 


the  formulas  k 


and? 


But  it  is  not  necessary  to  set  the  aperture  exactly  in  the  apex  of 
the  parabola,  it  may  be  placed  in  any  other  point  of  the  parabolic 
arc,  provided  that  the  axis  of  the  aperture  is  tangential  to  the  para- 
bola (Vol.  II.  §  93). 

A  third  method  of  laying  on  the  water,  consists  in  a  penstock 
with  guide-curves,  or  buckets,  Fig.  219.  This  arrangement  is  par- 
ticularly applicable  when  the  water  in  the  pentrough  is  subject  to 
great  variation  of  level.  The  apparatus  shown  in  Fig.  219  consists 
of  two  sluice-boards  A  and  B,  each  of  which  can  be  separately  ad- 
justed, and  thus,  not  only  the  head,  but  also  the  orifice  of  discharge, 
is  regulated.  It  is  not  possible  to  lay  the  water  on  to  the  wheel 


202 


CONSTRUCTION  OF  THE  CURB. 


tangentially  by  means  of  the  guide-buckets  DE ;  but  we  can  approxi- 
mate to  within  20  to  30  degrees  of  this.  The  water  flows  out  between 
the  guides,  according  to  the  law  for  the  discharge  through  short  addi 


tional  tubes,  and,  therefore,  the 

Fig.  219. 


efficient  ^  may  be  taken  as  ,82, 
or  when  the  bottom  of 
the  sluice-board  is  accu- 
rately rounded  on  the 
inside  ^  =  0,90.  Hence 
the  co-efficient  of  resist- 
ance is  greater  in  this 
case  than  in  overfall 
sluices,  or  in  the  ordi- 
nary penstock.  Assum- 
ing n  =  0,85,  the  height 
necessary  for  producing 
the  velocity  c  is  : 
, 


/    1    \2    ^_ 
\  0,85  /   '  2g 


=  1,384  — ,  and  hence 

the  portion  of  the  total 
height  h  remaining  in 
water-arc  is:  ha=h — h. 


1,384 


In 


the  case  of  a  variable  supply  of  water,  the  arrangements  are  adapted 
to  the  average  supply,  by  laying  the  outer  end  M  of  the  centre 
guide-curve  at  the  height  A2  above  the  foot  of  the  wheel  F.  In 
order  to  place  all  the  guides  at  the  same  a/igle  of  circumference  as 
the  wheel,  the  normal  position  of  which  is  3  inches  from  the  guides, 
they  are  drawn  tangential  to  a  circle  KL,  concentric  with  the  wheel, 
the  position  of  which  is  determined  by  the  direction  EK  of  the  first 
guide-curve. 

§  107.  Construction  of  the  Curb  or  Mantle,  and  of  the  Wheel— 
The  mantle  or  curb  by  which  breast  wheels  are  inclosed,  in  order 
to  retain  the  water  in  them  as  long  as  possible,  is  formed  either  of 
masonry  or  of  wood,  and  sometimes  of  iron.  The  object  of  the 
curb  is  the  better  fulfilled  the  less  the  play  between  the  outer  edge 
of  the  buckets  and  the  cylindrical  surface  of  the  curb,  as  the  water 
escapes  by  whatever  free  space  is  left.  The  play  amounts  to  \  an 
inch  in  the  best  constructed  wheels;  but  it  is  not  unfrequently  as 
much  as  an  inch,  or  even  2  inches.  When  the  wheels  are  of  wood 
and  the  curb  likewise,  a  play  of  \  an  inch  is  an  insufficient  allowance, 
because  the  curb  is  apt  to  loose  its  symmetry,  and  then  friction 
between  the  buckets  and  the  curb  might  ensue.  For  iron  wheels 
and  stone  curbs,  the  chances  of  deformation  are  small,  and,  there- 
fore, a  very  small  amount  of  play  is  sufficient.  When  the  wheels 
fit  too  closely  in  their  mantles,  stray  pieces  of  wood  or  ice  floated 


CONSTRUCTION  OF  THE  CURB. 


203 


on  to  the  wheel  may  have  very  injurious  consequences.  It  is, 
therefore,  necessary  to  have  a  screen  in  front  of  the  sluices  to  keep 
back  all  stray  matters. 

Stone  euros  are  constructed  of  properly  selected  and  carefully 
dressed  stone,  or  of  brick,  either  being  laid  in  good  hydraulic  mortar 
or  cement.  Wooden  curbs  AE,  Fig.  220,  are  composed  of  beams 


Fig.  220. 


*tf,  D,  E,  of  larch  or  oak,  carefully  planked  with  deals  curved  as 
required.  The  bed  of  the  curb  is  inclosed  by  side  walls,  so  that 
lateral  escape  of  water  is  prevented.  If  the  water  can  flow  off  by 
the  tail-race  with  the  velocity  with  which  it  is  delivered  from  the 

Fig.  221. 


204  CONSTRUCTION  OF  THE  CURB. 

wheel,  then  the  curb  may  be  finished  flush  with  the  bottom  of  the 
race,  as  at  E,  Fig.  221 ;  but  if  the  water  flows  with  a  less  velocity, 
then  the  race  should  be  cut  out  deeper,  as  at  JIE,  Fig.  220,  so  as 
to  avoid  all  risk  of  back  water. 

The  construction  of  these  wheels  differs  essentially  from  that  of 
overshot  wheels,  in  respect  of  the  buckets  of  the  latter  forming  cells, 
whilst  in  the  former  these  are  mere  paddles  or  floats;  and  this  gives 
rise  to  a  different  mode  of  connecting  the  buckets  with  the  rim  of 
the  wheel.  The  Germans  distinguish  Staler ader  and  Strauberdder. 
The  former  have  shroudings  like  the  overshot  wheels,  to  support 
the  floats;  in  the  latter,  the  floats  or  buckets  are  chiefly  supported 
by  short  arms  or  cantilevers,  which  project  radially  from  the  circum- 
ference of  the  wheel.  Fig.  219  is  a  wheel  with  shrouding,  Figs. 
220  and  221  are  Strauber cider.  Fig.  221  is  a  wooden  wheel,  and 
Fig.  220  an  iron  wheel.  Very  narrow  float  wheels  have  only  a 
single  narrow  ring  by  which  the  floats  are  attached  on  to  cantilevers. 
In  wooden  wheels  the  supports  for  the  floats  are  passed  between  the 
two  sides  of  a  compound  ring  forming  the  shrouding.  In  iron  wheels, 
on  the  other  hand,  they  are  either  cast  in  one  piece  with  the  seg- 
ments of  the  wheel,  or  they  are  bolted  on  to  these.  The  buckets 
or  paddles  are  usually  of  wood,  and  are  nailed  or  screwed  to  the 
above-described  supports.  The  floor  of  the  wheel  is  placed  on  the 
outside  of  the  rings  or  shrouding,  and  does  not  close  the  wheel  com- 
pletely, slits  being  left  for  the  escape  of  the  air,  as  is  represented 
in  Fig.  215,  in  which  DE  is  the  wheel-paddle,  composed  of  two 
pieces,  EF  a  piece  of  the  bottom  of  the  wheel,  and  G  the  air-slit  or 
ventilator. 

§  108.  The  Mechanical  Effect  of  Curl  Wheels. — The  mechanical 
effect  produced  by  wheels  hung  in  a  curb  or  mantle  consists,  as  in 
overshot  wheels,  of  the  mechanical  effect  produced  by  the  impact, 
and  that  by  the  pressure  or  weight  of  water.  The  formula  repre- 
senting the  efficiency  of  each  is  the  same,  save  that  the  determination 
of  the  loss  of  water  requires  a  different  calculation,  inasmuch  as  in 
the  one  case  the  water  is  lost  by  a  gradual  emptying  of  the  buckets 
in  the  arc  of  discharge,  and,  in  the  case  now  in  question,  the  water 
escapes  through  the  space  necessarily  left  between  the  wheel  and 
the  curb.  We  have,  therefore,  to  determine  how,  and  in  what 
quantity,  the  water  escapes  through  this  space,  and  hence  deduce 
the  loss  of  effect  to  the  wheel.  If  we  put,  as  for  overshot  wheels, 
the  velocity  with  which  the  water  goes  on  to  the  wheel  at  the  divi- 
sion line  =  cv  the  velocity  of  the  wheel  in  this  line  or  circle  =  vv 
and  the  angle  cr  E  vv  Fig.  222,  between  the  directions  of  these  velo- 
cities =  /*,  then  the  effect  of  impact : 

=  (c.cos.  /*  — v,)?,  ^  Q  y< 

g 

If,  further,  GK,  the  difference  of  level  between  the  point  of  entrance 
£,  and  the  surface  of  the  water  in  the  tail-race,  GH  =  A,,  we  have 
(neglecting  loss  of  water  through  free  space)  the  effect  of  the  weight 
of  the  water  s±=  hl  Qy,  and  hence  the  total  effect  is,  as  before : 


CONSTRUCTION  OF  THE  CURB.  205 


L  =  Pv 


b 

Fig.  222. 


\  Q 


From  the  theoretical  effect  given  by  the  formula,  we  have  to  deduct 
the  loss  by  the  escape  of  water.     This  loss  affects  chiefly  the  effect 


of  the  water's  weight,  as  the  water  escapes  uninterruptedly,  as  any 
given  bucket  or  float  BD  descends  successively  to  the  position  -Bj-Dp 
B2D?,  &c.,  to  the  lowest  point  FL.  The  free  space  forms  an  orifice 
of  discharge  through  which  the  water  escapes  under  a  variable  head 
BE,  l^Ej,  B2Ey  If  we  put  e  =  the  width  of  the  wheel,  and  the 
breadth  of  the  free  space  =  «,  the  area  of  the  orifice  of  discharge 
=  se,  and  if  we  put  for  the  head  BE,  B^EV  &c.,  z,  zv  z3,  &c.,  and  <}> 
for  the  co-efficient  of  discharge,  then  the  quantity  escaping  in  any 
instant  of  time  t,  through  the  free  space  : 


If  n  be  the  number  of  positions  of  a  float  assumed,  the  mean  escape 
from  each  cell  : 

=  ,  e  ,  ( 
or  for  one  second  : 


But  the  whole  of  the  cells,  or  the  water-arc,  corresponds  to  the  fall 
hv  and,  hence,  the  loss  of  mechanical  effect  may  be_put: 


There  is  a  slight  loss  of  water  on  each  side  of  the  wheel,  as  there 
must  be  a  free  space  of  1  to  2  inches  here.     If  we  put  arcs  BO, 

VOL.  II.  —  18 


206  FORMULA  FOR  TOTAL  EFFECT. 

BV0V  &c.,  of  the  curb  covered  with  water,  equal  to  lv  ?2,  &c.,  then 
the  water  escaping  by  the  sides,  as  it  were  through  a  series  of  notches 
or  small  weirs  : 


=  2  .  |  *  s  I,  x/2^,  2  .  f  «?>  s  12  v/%,  &c., 
and,  therefore,  the  corresponding  loss  of  mechanical  effect  : 


§  109.  Losses.  —  There  is  a  still  further  loss  of  effect,  when  the 
surface  of  the  water  in  the  lowest  cell  does 
F'g-  223-  _  ^  not  correspond  with  the  surface  of  the  tail- 
race,  as  represented  in  Fig.  223.  For  in 
this  case  the  water  flows  from  the  cell  BD 
D^BV  as  soon  as  the  float  BD  passes  the 
end  of  the  curb  jP,  and  acquires  a  velocity 
due  to  the  height  FM=  A2,  in  addition  to 
the  velocity  v  of  the  wheel.  This  height 
h2  is  variable,  but  its  mean  value  is  evi- 
dently £  A2,  and,  therefore,  the  head  to 
which  the  velocity  of  the  water  flowing  from  the  wheel  is  due  is 

not  —  ,  but  —  +  J  h2.     We  have  already  deducted  the  loss  due  to 

9         J! 
the  height  —  in  estimating  for  impact,  and  we  have,  therefore,  only 

\  Jiz  Q  y  to  deduct  from  the  effect  found.  From  this  we  see,  that  a 
sudden  fall  from  the  end  of  the  circle  should  be  adopted  only  in  cases 
where  back-water  is  to  be  feared. 

There  are  still  other  sources  of  loss  of  effect  in  breast  wheels, 
such  as  the  friction  of  the  water  on  the  curb,  and  the  resistance  of 
the  air  to  the  motion  of  the  floats  ;  but  these  are  comparatively  of 
slight  importance. 

§  110.  Formula  for  Total  Effect.  —  We  shall  now  give  the  formula 
for  the  total  effect  of  breast  wheels,  leaving  out  of  consideration 
the  loss  of  effect  by  escape  of  water  at  the  sides,  as  also  the  loss 
from  friction  of  the  water  and  resistance  of  the  air  ;  but  allowing 
for  the  escape  through  the  free  space  between  the  wheel  and  curb, 
and  for  the  friction  of  the  gudgeons.  The  formula  will  then  stand 
thus: 

7-       T,        (c.  cos.  u  —  v,}v.  n      ,    7    r.  7  j,  ~  r 

L  =  Pv  =  1-i  -  C  -  y-i  Qy+^Qy  —  <j>esw>Aiy  —f  G-v, 

g  a 

in  which  w  is  substituted  for  the  mean  velocity  of  discharge: 


and  r  for  the  radius  of  the  gudgeons.     Hence: 

L  -rk  **-*-*i)  **  +  hl(i-*i^}m\  Q  7  -r-fa  v. 

L  9  Q    /J  a 

or,  introducing  s  the  co-efficient  of  the  bucket's  filling,  generally 

=  i,  and  —  =  -  ,  we  have: 
Q       e  a  v 


FOKMULA  FOE  TOTAL  EFFECT.  207 


j  .  r(«.^>.  i—  .)«.  + 

L  ^  1 

If  we  put  the  total  fall,  measured  from  the  surface  of  the  water  in 
the  pentrough  to  the  surface  of  the  tail-race  =  h,  then,  instead  of  hv 
we  may  introduce  h  —  1,1  .  ^l_,  and  then: 


In  order  to  find  the  value  of  the  velocity  of  entrance  cv  for  which 
the  effect  is  a  maximum,  we  have  only  to  consider  when  the  expres- 
sion 


is  a  maximum.    Putting  -         '  **  —  =  k,  then  the  expression 

i,l(i_t£5) 

\  edvj 

to  be  made  a  maximum  becomes  kc^  —  c*.  But  we  know  from  Vol. 
I.  §  386,  that  this  becomes  a  maximum  for  ct  =.  -  ,  and,  therefore, 
it  is  evident  that  the  effect  will  be  a  maximum  when  the  velocity  of 
entrance  of  the  water  cl  =  -  v*  C08'J^  __ 

1,1(1  -"2) 

V         tdvj 

If,  from  the  necessarily  small  value  of  /i,  we  put  cos.  A*  =  1,  and 
assume  also,  that  there  is  no  loss  in  the  discharge  from  the  sluice, 

then  c.  =  -  ^  -  ;  and  hence  we  perceive  that  the  velocity  of  en- 
j  _  »  *w 

t  d  vl 

trance  must  be  made  greater  than  the  velocity  at  the  circumference 
of  the  wheel,  and  this  so  much  the  more  as  the  free  space  s  is  greater. 
The  loss  by  escape  of  water  is  not,  as  an  average,  more  than  10  to 

15  per  cent.,  or  *JL^.  _.  J    to  33<j,  and,  therefore,  the  velocity  of 

*  avt 

entrance  of  the  water  cl  =  y*  v1  to  f  %  vr  In  practice,  however,  ct 
is  made  =  2  vv  or  the  wheel  revolves  with  half  the  velocity  the  water 
has  acquired  at  entering  the  wheel,  because  in  this  way  the  loss  is 
not  much,  and  the  water's  entrance  is  more  easily  regulated. 

If  we  introduce  c^  cos.  p.  =  2  vv  or  v^  =  £  c:  cos.  p,  into  the  equa- 
tion above  found,  we  get: 


cos.  v?         /  g 

From  the  factor  (l  —  *8W\.  we  learn  that  the  maximum  effect 
\         «  d  v/ 


208  EFFICIENCY  OF  BREAST  WHEELS. 

does  not  take  place  here  when  v  =  0;  for  even  when  v  =  -  —  —  the 

«  a 

whole  effect  of  the  water's  weight  is  lost  by  the  water  escaping 
through  the  free  space. 

§  111.  Efficiency  of  Breast  Wheels.  —  Morin  has  made  a  number 
of  experiments  on  breast  wheels  of  good  construction.  He  has 
compared  his  experimental  results  with  those  of  the  theoretical 
formula  : 


9 

and  has  found  a  tolerable  agreement  between  the  two  to  subsist. 
when  the  formula  is  corrected  by  a  co-efficient  *,  or  if  we  put: 


One  wheel  on  which  Morin  experimented  was  of  cast  iron,  with 
wooden  buckets  placed  obliquely  to  the  sluice,  and  turning  in  a 
close-fitting  iron  curb.  The  wheel  was  21  feet  4  inches  in  diameter, 
and  4'  —  10"  wide  ;  the  fall  was  5'  —  6",  there  were  50  floats,  and 
the  velocity  of  revolution  was  from  3'  —  4"  to  nearly  8  feet  per 
second,  the  velocity  of  the  water  from  a  well-constructed  sluice 
being  from  9  feet  2"  to  10  feet  6".  The  co-efficient  *  was  found 
to  be  about  0,75,  and  the  efficiency,  including  the  friction  of  the 
axles,  nearly  0,60.  A  second  iron  wheel,  experimented  upon  by 
M.  Morin,  was  hung  in  a  well-fitting  sandstone  curb.  It  was  13 
feet  diameter,  and  13  feet  wide.  There  were  32  floats,  the  fall 
being  6  feet  6".  So  long  as  the  speed  of  the  wheel  did  not  differ 
more  than  from  47  to  100  per  cent,  from  that  of  the  water  entering 
it,  that  is  within  speeds  varying  from  1'  —  8"  to  6  feet,  the  co-effi- 
cient x  remained  nearly  constant,  viz.:  =  0,788,  and  the  efficiency 
of  the  wheel  was  =  0,70.  A  third  wheel  was  almost  entirely  of 
wood,  and  hung  in  a  close-fitting  curb.  Its  height  was  20  feet,  and 
it  had  40  floats.  Worked  with  a  common  sluice,  the  co-efficient  x 
=  0,792,  and  with  an  overfall  sluice  this  rose  to  0,809.  The  effi- 
ciency, however,  was  in  the  first  case  only  0,54,  and  in  the  second 
0,67.  If  from  these  results  we  adopt  a  mean  value,  we  get  for 
breast  wheels  with  penstock  sluice  : 

i-  0,77 


and  for  those  with  overfall  sluice  : 

L  =  0,80  /(*«*•*-»)»      h  \  Q 

V          ff  / 

from  which,  however,  the  mechanical  effect  consumed  by  the  friction 
of  the  axles  has  to  be  deducted.  The  greater  efficiency  of  the 
overfall-sluiced  wheels  arises  from  the  water  entering  more  slowly 
than  in  the  case  of  the  penstock,  and,  hence,  there  is  no  loss  by 
impact. 

It  follows,  besides,  from  Morin's  experiments,  that  the  efficiency 
diminishes  if  the  water  fills  more  than  from  J  to  f  of  the  space  be- 


EFFICIENCY  OF  BREAST  WHEELS.  209 

tween  the  floats,  and  that  the  efficiency  does  not  vary  much  for  varia- 
tions of  the  angular  velocity  of  the  wheel  from  1'  —  8"  to  6' 6" 

per  second. 

Egen  made  experiments  on  a  breast  wheel  23  feet  in  diameter, 
and  4£  feet  wide.  There  were  two  peculiarities  in  this  wheel.  The 
69  well-ventilated  buckets,  were  constructed  exactly  as  in  overshot 
wheels ;  and  the  sluice  was  in  two  divisions,  of  which,  according  to 
the  state  of  supply  of  water,  the  upper  or  under  one  could  be  drawn. 
Although  the  mantle  fitted  very  closely,  the  efficiency  of  this  wheel 
was,  at  best,  only  0,52,  and  as  an  average,  with  6  cubic  feet  water 
per  second,  and  4  revolutions  per  minute,  the  efficiency  was  only 
0,48. 

Experiments  with  a  breast  wheel  are  described  in  the  "Bulletin 
de  la  SocietS  Indust.  de  Mulhouse,  L.  XVIII."  The  wheel  was  of 
wood,  5  metres  or  16,4  feet  in  diameter,  and  13  feet  wide,  made  in 
three  divisions  on  2  centre  shroudings.  The  curb  started  from  a 
parabolic  saddle-beam  8  inches  in  height,  and  the  water  was  laid  on 
by  an  overfall  sluice  8  inches  high.  Thus  the  velocity  of  the  water 
was  about  8,8  feet,  and  the  angular  velocity  of  the  wheel  from  5 
feet  to  6'  —  6".  The  buckets  were  filled  from  J  to  |,  and  the  effi- 
ciency increased  as  the  buckets  were  more  filled.  When  the 
buckets  were  quite  filled,  the  efficiency  was  0,80  ;  when  half-filled, 
it  was  0,73 ;  and  with  less  water,  it  was  only  0,52.  The  experi- 
/•ments  on  the  efficiency  of  the  wheel  for  different  degrees  of  filling  of 
the  buckets,  were  easily  and  precisely  made  in  this  case,  from  the 
circumstance  that  the  water  could  be  laid  on  each  division  of  the 
wheel  separately.* 

Example.  Required  the  calculated  proportions  of  a  breast  wheel,  being  given  Q=  15 
cubic  feet  per  second,  h  =  8$  feet,  and  the  velocity  of  revolution  5  feet.  We  shall  assume 
the  depths  of  the  floats  or  of  the  shrouding  to  be  1  foot,  and  suppose  the  buckets  to  be 

2O         30 

filled  to  $  their  contents.     The  width  of  the  wheel  is,  hence,  e  =  _  = —  6  feet. 

dv         1.5 

Assume  also  that  the  water  enters  with  double  the  velocity  of  rotation,  then  c=2  .  5=  10 
feet,  and  the  fall  required  to  generate  the  velocity 

At  =  1,1  —=  1,1  .  0,00155  .  100  =  1,705  feet. 

2g 

Deducting  this  impact  fall  from  the  total  fall,  there  remains  for  the  height  of  the  curb, 
or  for  the  fall  during  which  the  water's  weight  alone  acts,  A"  =  h  —  A,  =  8,5 — 1,705 
=  6,795  feet.  We  shall  adopt  a  large  wheel,  that  the  water  may  not  fall  too  high  into 
it.  Making  the  radius  a  =12  feet,  and  the  radius  of  the  division  line  =  11,5  feet. 
The  water  revolving  with  the  velocity  of  the  wheel,  we  shall  suppose  to  be  carried  to 
the  bottom  of  the  curb,  as  represented  in  Fig.  224.  The  central  angle  a  of  the  curb 
EG,  or  the  angle  by  which  the  points  of  entrance  of  the  water  E  deviates  from  lowest 
point  F,  is 

cos,  a  =  1  —  **  =  1  —  6'795  =  0,4092, 

ot  11,5 

and,  hence,  a,  =  65°,  507.  We  shall  assume  that  the  direction  Ect  of  the  water 
deviates  20  degrees  from  the  direction  Evt  of  the  wheel's  motion  at  the  division 


*  [For  the  efficiency  of  breast  wheels,  with  elbow  and  centre  buckets,  see  above  (p. 
182,  note).  The  Franklin  Institute  committee  found,  with  a  15  feet  wheel  and  elbow 
buckets,  taking  the  water  10,46  feet  above  the  bottom  of  the  wheel,  an  efficiency  from 
612  to  677,  and  laying  in  on  7  feet  from  the  bottom,  the  efficiency  varied,  with  different 
heads,  from  ,568  to  ,631.— AM.  ED.] 

18* 


210 


EFFICIENCY  OF  BREAST  WHEELS. 


circle,  and  refer  the  velocity  of  5  feet,  in  like  manner,  to  the  division  or  pitch  circle. 
We  then  have  the  co-ordinates  of  the  summit  of  the  parabolic  saddle,  JlM  =  k  = 
r2M».  (45°50')a  _  Q  an(J  ME  _  l  _  CJL  gin  91o  4/  _  j  55  feet  according  to  which 

2g  2g 

dimensions  the  construction  of  Fig.  224  has  been  carried  out.     The  height  of  the  water 

Fig.  224. 


.aR  above  the  sill  is  fe)  —  fc=  1,705  —  0,S  =  0,905,  and  if  we  put  the  height  of  the  orifice 


0,905  - 


15 


t*e    /'2£  f  0,905  —  - 
N       \  2, 

0.35 


0,9  .  6  .  8,02 


o,905  —  f. 


14,43  J  0,905  —  | 


:  and,  hence,  x  =  0  .  4  feet. 


The  theoretical  effect  of  this  wheel  is  Z  = 
/(*«•.*-•)* 


|          Q  y  = 


+  6,795)  .  15  .  62,25  =  7244  feet 


Ibs.,  and  the  whole  available  effect  is  8,5  X  933  feet  Ibs.  =  7930  feet  Ibs.  We  have 
now  to  deduct  the  loss  by  the  escape  of  water  through  the  free  space  between  the  curb 
and  the  wheel.  Assuming  the  play  to  be  1  inch  =  ^?  feet,  then  the  area  of  the  slit  by 
which  the  water  can  escape  is  r'?  .  6  =£  square  feet.  In  order  now  to  find  the  mean 
velocity  w,  with  which  the  water  passes  through  this  aperture,  the  height  of  curb  KG  is 
to  be  divided  into  6  equal  parts,  and  the  position  of  the  buckets  for  each  point  so  found, 
accurately  delineated,  as  is  done  in  Fig.  224,  and  the  heads  or  pressures  measured. 
Commencing  at  the  top,  we  have  :  z  =  0,80,  z,  =  0,80,  za  =  0,80,  *3  =  0,80,  z4  =  0,67. 
s5  =  0,48  and  z6  =  0.  From  this  we  have  the  mean  of  the  square  roots  of  these 
quantities  = 

$.0.894  +  0,894+0,894  +  0,818+0,693+0 

6 

and,  hence,  the  mean  velocity  of  escape  of  the  water  =  8,02  X  0,7736  =  6,188.  The 
mechanical  effect  corresponding  to  this  is  L,  =  $esw  A,  y,  in  which  f  =  0,7  the  co- 
efficient of  discharge,  therefore, 

i  =  0,7.i.  6,188  X  6,79  X  62,25  =  916  feet  Ibs. 

The  loss  by  escape  at  the  sides  of  the  wheel  may  be  calculated  by  the  formula  given  § 
108.  It  will  be  found  =  180  feet  Ibs.  So  that  the  total  loss  by  the  escape  of  water 
=  1096  feet  Ibs  ;  deducting  this  from  7244  feet  Ibs.,  there  remain  6148  feet  Ibs. 


UNDERSHOT  WHEELS. 


211 


effective.  The  escape  of  water  in  this  wheel  we  see  involves  a  loss  of  15  per  cent,  of 
the  mechanical  effect  of  the  fall.  By  the  friction  of  the  water  and  the  resistance  of  the 
water,  the  loss  is  about  160  feet  Ibs.,  or  about  2$  per  cent.  There  remains,  therefore, 
5988  feet  Ibs. 


If  now  we  take  the  weight  of  the  wheel  G  = 


,  the  ratio  of  rilling  the  buckets, 


30     5        2*5  5988 

being  i,  we  have  u  =  -  1_  —  _  _  4  and  L  =  -  -_  1  1  Ibs.  feet  nearly  :  then,  the 
•an         2  if  550 

SOOO  v   12 
weight  of  the  wheel  =          .*        =  1650  Ibs.     Hence,  the  radius  of  the  gudgeons 

r  =  0,002  ^8250  =  0,182  feet,  and  from  this  we  get  the  mechanical  effect  absorbed  by 

0.   19       n,       1CKnn      r  ,  o*    *._    «L  ing   thig    further  de. 

10,6  feet  Ibs.,  and,  lastly,  the  efficiency  of  the 


friction  =  -  fG  v  =     '       ..  0,1  .  16500  .  5  =  136  feet  Ibs. 

a  11.5 

duction,  there  remains  5852  feet  Ibs. 

,      ,  5852        n „. 

wheel  i)  = ==  0,74. 

7930 


Fig.  225. 


§  112.  Undershot  Wheels. — Undershot  wheels  usually  hang  in  a 
channel  made  to  fit  as  closely  to  the  wheel  as  possible,  so  that  water 
may  not  escape  without  producing  its  effect.  Hence,  the  application 
of  a  channel  having  a  curb  concentric  with  the  wheel  is  considered 
better  than  a  straight  channel  tangential  to  the  wheel.  The  curb 
allows  of  some  of  the  effect  of  the 
weight  of  the  water  being  availed  of. 
The  calculation  for  such  a  wheel 
as  is  represented  in  Fig.  225,  when 
*>  the  curb  JIB  embraces  3  to  4  floats 
at  least,  is  identical  with  these  for 
the  breast  wheels  last  considered. 
The  rules  for  construction  of  under- 
shot wheels,  correspond  too  with 
those  for  breast  wheels.  The  floats 
are  usually  put  in  radially ;  but 
sometimes  they  are  inclined  up- 
wards towards  the  sluice,  that  they 
may  carry  no  water  up  with  them  on  the  opposite  side.  These 
floats  are  not  unfrequently  composed  of  two  equal  pieces  J1D  and 
BD,  Fig.  226,  so  that  the 
angle  ADB  =  100°  to  120°. 
This  arrangement  allows  of 
ample  openings  being  left  in 
the  flooring  of  the  wheel,  with- 
out fear  of  the  water  flowing 
through  the  sluice.  The  cells 
or  buckets  are  allowed  to  fill 
from  one-half  to  two-thirds  of 
their  capacity,  or  *  =  J  to  f . 
To  prevent  overflow  of  the 

water  inwards,  or,  in  order  to  have  greater  capacity,  the  depth  of 
the  wheel,  t.  e.,  of  the  shrouding,  is  made  from  15  to  18  inches. 
The  laying  on  the  water  tangentially  is  more  rarely  done  than  in 
breast  wheels.  The  sluice-board  is  inclined  in  order  that  the  sluice- 


Fig.  226. 


212  UNDERSHOT  WHEELS. 

aperture  may  lie  as  close  to  the  wheel  as  possible.  The  lower  edge 
should  be  rounded  off  to  prevent  partial  contraction  of  the  vein  of 
water. 

§  113.  The  effect  of  undershot  wheels  is  less  than  that  of  breast 
wheels,  the  fall  available  as  weight  being  greater  in  the  latter.  The 
half  of  the  fall  is  necessarily  lost  when  it  acts  by  impact,  whereas 
the  loss  by  escape  of  water  acting  by  its  weight  on  those  wheels  does 
not  amount  to  \  of  the  whole.  Experiment  has  satisfactorily  estab- 
lished this.  The  wheel  with  which  Morin  experimented  was  19'. 6" 
in  diameter,  5J  feet  wide,  and  had  36  radial  floats.  The  sluice  was 
inclined  at  an  angle  of  34J°  to  the  horizon,  and  the  sluice-aperture 
was  fl\  feet  back  from  the  commencement  of  the  curved  course. 
The  total  fall  was  6' — 3",  and  the  head  on  the  sluice-aperture  4' —  7". 
There  was,  therefore,  a  fall  of  about  1'  —  8"  through  which  the 
water's  weight  acted.  The  velocity  of  the  circumference  of  the 
wheel  was  from  6'  —  6"  to  13'  —  0"  :  and  the  velocity  of  the  water 

on  reaching  the  wheel  from  16  to  18  feet.     As  long  as  -  did  not  ex- 

c 

ceed  0.63,  the  efficiency  ^  was  0.41  as  a  mean:  but  when  -  varied 

c 

between  the  limits  0.5  and  0.8,  then  the  mean  efficiency  7  was 
only  0.33. 

Retaining  our  former  notation,  we  have,  for  the  effect  of  this 
wheel,  exclusive  of  friction  of  gudgeons, 


in  the  first  case;  and 

jPW_AtfO(£=*L!+-A|)4y 

*       9  ' 

in  the  second. 

A  second  wheel  with  which  Morin  experimented,  was  about  13 
feet  high,  2'  —  8"  wide,  11,8  inches  deep,  and  had  24  floats.  The 
water  was  laid  on  by  a  vertical  sluice,  and  reached  the  wheel  through 
a  straight  course  2'  —  8"  long.  This  channel  and  the  curb  were  of 
sandstone,  and  the  free  space  left  amounted  to  only  0,2  of  an  inch. 
The  mean  fall  was  3  feet.  The  head  of  water  on  the  sluice-aperture 
varied  from  6  to  18  inches.  Experiments  were  made  at  various 
velocities  of  rotation.  For  small  velocities  the  efficiency  was  very 
small.  For  the  mean  velocity  of  5  feet  it  was  a  maximum,  and, 
when  the  velocity  of  the  water's  arrival  on  the  wheel  was  not  much 
different  from  this,  a  maximum  efficiency  0,49  was  obtained.  For 

ratios  of  velocities  -  =  i  and  -  =  f ,  the  mean  value  of*  was  found 

to  be,  as  for  the  former  wheel  0,74.     Hence 

Pv  =  0,74  /(c  —  v)v  +  ^  \  Q  ^  .g  the  formula  for  ^  cage  algo< 

*        9  ' 

Morin  puts  together  the  results  of  his  experiments  on  wheels  con- 


WHEELS  IN  STRAIGHT  COURSES. 


213 


fined  in  mantles  or  curbs  as  follows.  Wheels  in  which  A:  =  %h 
x  =  0,40  to  0,45.  When  A,  =  f  A,  *  =  0,42  to  0,49.  When  h,  =  f  A, 
*  =  0,47,  and  when  h,  =  f  £,  *  =  0,55. 

Example.  Required,  the  effect  of  an  undershot  wheel,  15  feet  in  diameter,  and  making 
8  revolutions  per  minute.     The  fall  4  feet,  and  the  quantity  of  water  20  cubic  feet  per 

second,  v  =.  ™  u  a  =  *  '  8  '  15  =  6,283  feet,  and  supposing  the  velocity  of  the  water 
to  be  double  this:  then  the  pressure  of  the  water  in  front  of  the  sluice,  or  what  we  have 
termed  the  impact-fall  =  1,1  —  =  1,1  x  0,0155  x  12,56'  =  2,689  feet,  and  there  there- 
fore remains  as  fall,  through  which  the  water  acts  by  its  weight,  A,  =  4  —  2,689  ^  1,31 1 
feet,  and  hence  the  theoretical  effect  =  (0,031  .  6,283*+  1,311)  20  .  62,5  =  (1,263 


+  1,311)  1245  =  3264  feet  Ibs. 


In  this  case,  A,  =  h 


0,33  h,  and,   therefore, 
13708  feet 


the  co-efficient  x,  may  be  taken  0,42,  and  hence  the  effect  Z  =  0,42  .  3264 
Ibs.  from  which,  however,  the  gudgeon-friction  has  to  be  deducted. 

§  114.  Wheels  in  Straight  Courses. — When  the  undershot  wheel 
is  hung  in  a  straight  course,  the  effect  is  a  minimum;  because  the 
water  produces  its  effect  by  impact  alone,  and  a  considerable  quan- 
tity escapes  unused.  These  wheels  are  only  adopted  for  falls  of  less 
than  4  feet,  and  where  water  power  is  of  value  the  Poncelet-wheel, 
or  turbines,  are  now  invarially  preferred.  They  are  made  from  12 
to  24  feet  in  diameter,  with  24  to  48  floats,  usually  radial,  but 
sometimes  placed  with  a  slight  inclination  towards  the  sluice.  The 
„  breadth  or  depth  of  the  floats  should  be  made  about  three  times  the 
thickness  of  the  layer  of  water  coming  through  the  sluice,  because 
the  water  in  contact  with  the  wheel  retains  only  35  to  40  per  cent, 
of  the  velocity  of  the  water  before  impact,  when  the  greatest  effect 
is  produced;  and,  therefore,  the  stream  of  water  flowing  along  as 
the  wheel  revolves  is  2J  to  3  times  the  thickness  of  the  water  com- 
ing from  the  sluice.  The  depth  of  the  sluice-aperture  is  usually  4 
to  6  inches,  and  thus  the  floats  are  made  from  12  to  18  inches  deep 
for  the  above  reason.  The  straight  course  in  which  undershot 
wheels  are  suspended  may  be  either  horizontal  as  in  AB,  Fig.  227, 

Fig.  227. 


or  inclined,  as  in  Fig.  228.     That  as  little  water  as  possible  may 
escape  unemployed,  the  space  between  the  wheel  and  the  course 


214  USEFUL  EFFECT  OF  UNDERSHOT  WHEELS. 

must  be  reduced  to  1  or  2  inches  at  most.  And  hence,  it  is  better 
to  lay  the  course  with  a  slight  curvature,  the  floats  being  made  so 
numerous,  that  there  are  always  4  or  5  floats  submerged. 

Fig.  228. 


The  penstock  is  set  with  an  inclination  to  bring  the  orifice  of 
discharge  as  near  to  the  wheel  as  possible,  and  to  avoid  contraction. 
To  prevent  back-water,  the  course  is  made  to  drop  suddenly  some 
inches,  at  the  point  where  the  water  quits  the  wheel.  Besides  this 
provision,  arrangements  for  raising  and  depressing  the  wheel,  or  the 
water-course,  are  adopted. 

Fig.  227  represents  a  lift  for  the  wheel  (called  in  German  Zieh- 
panster).  The  axle  M  of  the  lever  MD  coincides  with  that  of  the 
wheel,  so  that  the  connection  between  the  driving  wheel  and  pinion 
may  not  be  altered  in  raising  or  depressing  the  water  wheel.  All 
these  arrangements  are,  however,  rendered  unnecessary  by  the  adop- 
tion of  the  turbine,  instead  of  undershot  wheels,  in  all  cases  in  which 
the  water  is  liable  to  much  variation. 

§  115.  Useful  Effect  of  Undershot  Wheels.—  Experiments  on  the 
useful  effect  of  undershot  wheels,  with  straight  courses,  have  been 
made,  but  only  on  models,  by  De  Parcieux,  Bossut,  Smeaton, 


Lagerhjelm,  &c. 
Th 


e  experiments  of  Smeaton  and  Bossut  are  the  best.*  The  re- 
sults of  the  experiments  are  satisfactorily  in  agreement  with  each 
other,  and  confirm  the  theory.  The  mechanical  effect  evolved  by 
these  wheels  was  ascertained  in  all  the  experiments,  by  raising  a 
weight  by  means  of  a  cord  passed  round  the  axle  of  the  wheels. 
Smeaton's  experiments  were  made  with  a  small  wheel  75  inches  in 
circumference,  having  24  floats,  each  4  inches  wide,  and  3  inches 
deep.  The  general  conclusion  at  which  Smeaton  arrived  is,  that 

for  the  velocity  ratio  -  =  0,34  to  0,52,  the  maximum  useful  effect 
c 

amounts  to  0,165  to  0,25.  Bossut's  experiments  were  made  with 
a  wheel,  3  feet  in  diameter,  provided  with  48,  with  24,  and  with  12 
buckets,  5  inches  wide,  and  4  to  5  inches  deep.  Bossut  found,  as 

•  [See  foot  note  and  reference  next  page.  —  Ax.  ED.] 


PARTITION  OF  WATER  POWER.  215 

theory  indicates,  that  with  48  floats,  the  efficiency  is  greater  than 
with  24,  and  with  24  greater  than  with  12 ;  and  he  deduced  from 
his  experiments  that  about  25°  of  the  wheel's  circumference,  or 
32g55  .  48=  J3°,  or  more  than  3  floats  should  be  in  the  water  at  the 
same  time.  From  Bossut's  experiments  on  the  wheel  with  48  floats, 
a  somewhat  greater  efficiency  results  than  is  indicated  by  Smeaton's 
experiments,  and  this  may  probably  be  attributed  to  the  greater 
proportional  number  of  buckets  in  Bossut's  model.*  The  mean 
result  of  the  two  sets  of  experiments  gives  the  effect  of  such  wheels, 
friction  not  taken  into  account : 

£  =  0,61  (c~v)v  Qy  =  l,19(c  —  v)v  Qfeetlbs. 
g 

This  formula  will  only  apply  on  the  scale  of  practice,  when  the 
play  allowed  between  wheel  and  course  is  not  greater  than  1J  inch. 
Instead  of  Q  we  have  Fc,  in  which  F  is  the  arc  of  the  float  dipping 
into  the  water ;  and  hence  we  have  the  formula  given  by  Christian 
in  his  "MScanique  industrielle,"  substituting  0,76  for  0,61. 

L  =  0,76  Fy  .  (c~^  c  v  =  1,48  (c  —  v)  Fc  v  feet  Ibs. 

g 

From  the  experiments  extant,  it  follows  also,  that  the  maximum 
effect  is  produced  for  the  velocity  ratio  -  =  0,4,  as  indicated  by 

»v  theory.     For  greater  velocities  this  ratio  is  somewhat  less,  and  for 
large  bodies  of  water  the  ratio  is  somewhat  greater. 

§  116.  Partition  of  Water  Power. — A  given  fall  of  water  is  often 
divided  between  several  wheels,  not  only  because  a  single  wheel 
would  have  cumbrous  dimensions,  but  more  especially  for  the  sake 
of  working  different  machines  or  tools  independently,  avoiding  the 
coupling  connections  with  one  source  of  power.  The  question  may 
arise  as  to  a  division  of  height  of  fall,  or,  as  to  partition  of  the 
quantity  of  water.  As  a  general  rule,  we  may  assume  that  for 
wheels  on  which  the  water  acts  by  its  weight,  a  partition  of  the 
quantity  of  water,  and  for  wheels  on  which  the  water  acts  by  impact, 
a  partition  of  the  height  of  fall  is  to  be  preferred;  for  we  have  seen 
that  the  efficiency  of  overshot  wheels  of  great  diameter,  is  greater 
than  that  of  overshot  wheels  of  smaller  diameter,  or  even  than 
breast  wheels;  and,  on  the  other  hand,  it  is  manifest  that  the  loss 
of  effect  by  impact,  and  by  the  escape  of  water  through  the  wheels, 
is  less  when  these  wheels  are  placed  one  behind  the  other,  than  when 
they  are  put  side  by  side,  because  the  velocity  due  to  the  height 

corresponding  to  the  loss  of  effect  ^ — - — -  (Vol.  I.  §  387),  and  the 

ratio  —  of  the  free  space  to  the  depth  of  water,  is  less  than  in  the 

*i 
latter  case.     For  breast  wheels  in  a  curb,  on  which  the  water  acts 


*  [The  ratio  of  effect  to  power,  obtained  by  the  Committee  of  the  Franklin  Institute, 
was  found  to  vary  from  ,266  to  ,305,  and  the  average  is  set  down  at  ,285.  See  Journal 
Franklin  Institute,  3d  series,  Vol.  IL,  p.  2,  for  July  1841.— AM.  ED.] 


216 


FLOATING-MILL  WHEELS. 


by  its  weight  and  by  impact,  and  in  which  the  loss  of  water  depends 

mainly  on  — ,  there  is  no  general  rule  for  the  preference  of  one 

*. 

mode  of  partition  over  the  other,  and  the  circumstances  of  each  case 

must  determine  our  choice. 

§  117.  Floating-mill  Wheels. — Wheels  suspended  on  two  boats, 
or  barges,  conveniently  moored  in  a  river,  are  undershot  wheels 
without  curb  or  limited  course  of  any  kind.  These  wheels  are  sup- 
ported either  on  two  boats,  one  of  which  contains  the  mill  machinery, 
or  one  end  of  the  axle  rests  on  a  boat,  the  other  resting  on  piles 
driven  in  on  shore,  in  which  case  the  mill  machinery  is  kept  on 
shore. 

The  construction  of  boat-mill  wheels  differs  from  that  of  ordinary 
undershot  wheels,  inasmuch  as  they  have  no  shrouding,  the  floats 
being  attached  directly  to  the  arms.  These  wheels  are  made  from 
12  to  15  feet  in  diameter,  and  have  generally  only  6  or  7  floats, 
although  10  to  12  would  constitute  a  better  wheel.  The  floats  are 
made  long  and  very  broad,  that  they  may  catch  a  large  stream  of 
water,  for,  the  velocity  being  usually  small,  the  vis  viva  depends  in 
a  great  measure  on  the  mass.  Floats  of  6  to  18  feet  in  length,  and 
2  feet  to  30  inches  broad,  are  usual.  The  floats  are  inclined  at 
angles  of  from  10°  to  20°  to  the  radius,  and  dip  to  about  one-half 
their  breadth  into  the  stream. 

Fig.  229  represents  a  boat-mill  (Fr.  moulin  a  nef ;  Ger.  Schiff- 

Fig.  229. 


muJde).  A  being  the  mill-house  on  the  barge  -B,  and  C  a  wheel 
with  6  floats,  the  axle  of  which  passes  through  the  mill-house,  and 
projects  as  far  on  the  opposite  side  of  the  boat  as  the  one  seen  in 
the  figure  does  on  this.  The  mill  gear  is  within  the  house. 

The  effect  of  boat-mill  wheels  is  less  than  of  wheels  hung  in  a 
confined  course,  for  two  reasons,  viz. :  the  water  not  only  escapes  by 
the  sides,  and  under  the  floats,  but  a  considerable  quantity  passes 
through  the  wheel  without  coming  into  action,  from  the  small  num- 
ber of  buckets  that  dip  into  the  water. 


FLOATING-MILL  WHEELS.  217 

§  118.  The  theoretical  effect  of  a  freely  suspended  water  wheel 
may  be  represented,  as  for  undershot  wheels,  by  the  formula 


c  and  v  being  the  velocities  of  the  water  and  of  the  wheel,  and  F  the 
area  of  the  part  of  the  float  dipped,  neglecting  the  damming  up  of 
the  water  upon  it.  This  expression  has  to  be  multiplied  by  a  co- 
efficient allowing  for  the  loss  of  water. 

§  119.  Experiments  on  the  effect  of  these  wheels  have  been  made 
by  De  Parcieux,  Bossut,  and  Poncelet,  but  principally  on  models. 

Bossut's  model  wheel  was  3  feet  in  diameter,  had  24  floats,  6£ 
inches  wide,  dipping  4J  inches  into  the  water.  The  velocity  of 
the  water  was  6  feet  per  second.  The  result  of  these  experiments 
gives  /*  =  1,37  to  1,79  as  the  co-efficient,  by  which  the  formula 

L  =  (c~~v)vc  Fy  is  to  be  multiplied,  and  ^  =  0,877  to  0,706  as 

9 
the  co-efficient  for  the  formula  L  =  I       v ) v  c  p^  ^gee  p ' Aubuisson, 

"  Hydraulique,"  §  352).  The  limits  of  the  values  of  the  co-efficients 
are  nearer  each  other,  in  this  latter  case,  than  in  the  other,  which 
was  to  be  expected,  as,  from  the  number  of  buckets,  the  second  for- 
mula is  most  applicable.  The  number  of  buckets  should  be  such 
that  2  at  least  are  in  the  water,  and  then  the  latter  formula  with  the 
mean  co-efficient  p  =  0,8,  will  apply,  or, 

L  =  0,8(<?~  v">cvFy  =  1,55  (c  —  v)  cv  Ffeet  Ibs. 

9 

Poncelet's  observations,  made  on  three  boat-mill  wheels  on  the 
Rhone,  agree  with  this.  These  wheels  were  8  to  10  feet  long,  and 
the  floats  dipped  2'  —  3"  to  2'  —  9"  into  the  water  flowing  with  a 
velocity  of  from  4  to  6J  feet  per  second.  Poncelet  cites  an  experi- 
ment by  Boistard,  and  one  by  Christian,  both  of  which  confirm  the 
accuracy  of  this  formula. 

Bossut's  experiments,  in  exact  accordance  with  theory,  show  that 
the  maximum  effect  is  obtained  when  v  =  0,4  c,  and  Poncelet's  ex- 
periments on  the  Rhone  boat  wheels  also  give :  -  =  0,4. 

c 

Introducing  v  =  0,4  c  into  the  above  formula,  the  useful  effect 
becomes : 

L  =.  0,8  °'6  '  °'4  ^  Fy  =  0,192  ^-  Fy  =  0,384  —  Q  y, 

ff  9  *9 

and,  hence,  the  efficiency  q  =  0,384. 

De  Parcieux's  experiments  were  specially  directed  to  ascertaining 
the  best  position  for  the  floats.  The  result  was  that  an  inclination 
of  60°  to  the  stream  is  the  best. 

Remark.  There  has  long  been  a  doubt  as  to  which  of  the  two  formulas 
J—  f*  (C  —  VTV  fy  an(j  L  __(*i  (c  —  v)cv  jry  is  the  more  correct.    The  one  is  known 

as  Parent's  formula,  the  other  as  Bordas.    Now,  although  for  a  wheel  in  an  unconfined 
VOL.  II.— 19 


218  PONCKLET'S  WHEELS. 

stream  acting  on  the  floats,  all  the  water  going  through  the  wheel  does  not  assume  the 
velocity  of  the  floats,  yet,  considering  the  great  extent  of  the  floats'  surface,  it  may  cer- 
tainly be  presumed  that  the  greater  part  of  the  water  on  impinging,  takes  the  velocity  of 
the  floats,  and,  hence,  the  greater  accordance  between  experiments  in  Borda's  formula  is 
explained.  Parent's  formula  is  founded  on  the  assumption  that  the  impact  is  propor- 
tional to  the  height  due  to  the  relative  velocity  c—v.  (Compare  Vol.  I.  §  392,  where  the 

force  of  impact  is  given  =1,86  —  F  y  when  v  =  0. ) 
^S 

§  120.  Poncelet's  Wheels. — If  the  floats  of  undershot  wheels  be 
curved  so  that  the  stream  of  water  runs  along  the  concave  side, 
pressing  upon  it  without  impact,  the  effect  produced  is  greater  than 
when  the  water  impinges  at  nearly  right  angles  against  straight 
buckets. 

Poncelet  introduced  these  wheels.  They  are  of  very  advantageous 
application  for  low  falls  under  6  feet,  because  their  effect  is  much 
greater  than  that  of  undershot  wheels  with  or  without  a  curb.  For 
greater  falls,  breast  wheels  with  a  well-formed  circle  excel  them, 
and  as  their  construction  is  more  difficult,  they  are  not  applied  for 
greater  falls  than  6  feet.  Poncelet  has  treated  of  these  wheels  in  a 
special  work,  entitled  "M6moire  sur  leg  Roues  hydrauliques  d  aubes 
courbes,  mues  par-dessous,  Metz,  1827."  Fig.  230  represents  the 

Fig.  230. 


general  arrangement  of  these  wheels:  JJ?is  an  inclined  sluice-board; 
JIB  is  the  stream  of  water  entering  the  wheel  at  the  buckets  BD 
and  BJ)r  -FGr  is  the  surface  of  the  tail-race.  In  order  that  nearly 
all  the  water  may  come  into  action,  the  wheel  must  have  very  little 
play  in  the  water-course,  and  to  prevent  partial  contraction,  the 
under  side  of  the  sluice-board  is  rounded  off:  also,  to  prevent  loss  of 
vis  viva  by  friction  in  the  channel,  the  aperture  of  the  sluice  is 
brought  clos'e  to  the  wheel.  The  first  part  of  the  course  AB  is 
inclined  at  1'5  to  TJ5.  The  remainder  of  the  course,  which  embraces 
the  length  occupied  by  three  buckets  at  least,  is  accurately  curved 
concentrically  with  the  wheel,  and  at  the  end  of  it,  a  sudden  dip  of 
6  inches  is  made,  and  the  tail-race  should  also  be  widened  to  guard 
against  any  liability  to  back-water  on  the  wheel.  Poncelet  wheels 


PONCELET'S  WHEELS. 


219 


have  been  constructed  from  10  to  20  feet  in  diameter,  and  with  32 
to  48  floats  of  sheet  iron  or  of  wood.  Wooden  floats  are  composed 
of  staves,  like  a  barrel,  the  outer  edge  being  sharpened,  or  provided 
with  a  sheet  iron  edge  piece.  Sheet  iron  floats  are,  however,  much 
more  suitable,  as  good  construction  is  an  essential  feature  in  this 
wheel.  The  sluice  is  not  drawn  more  than  1  foot  in  any  case,  and 
for  falls  of  5  to  6  feet,  an  aperture  6  inches  high,  or  less,  is  ar- 
ranged for. 

§  121.   Theory  of  Poncelet's  Wheels. — To  obtain  the  maximum 
elFect  from  these  wheels,  the  water  must  go  on  to  the  buckets  with- 
out impact.      If  Ac  =  c  (Fig. 
231)   be    the    velocity   of    the  Fig.  231. 

water  going  on  to  the  wheel, 
and  Av  =  v,  the  velocity  of  the 
periphery  of  the  wheel,  we  then 
have  in  the  side  Ac1  =  cl  of  the 
parallelogram  A  v  c  cv  the  ve- 
locity of  the  water  in  reference 
to  the  wheel,  both  in  magnitude 
and  direction.  If,  therefore, 
we  put  the  curved  float  AK 
tangential  to  Ac^  the  water  will 
begin  to  ascend  along  it  without 
*>  the  least  shock,  with  the  velo- 
city cr  If  we  put  the  angle 

c  A  v  by  which  the  direction  of  the  water  deviates  from  that  of  the 
circumference  of  the  wheel,  or  the  tangent  Av  =  8,  we  have  for  the 
relative  velocity  of  the  water  beginning  its  ascent  on  the  floats 
=  cl  v/e2  -1-  v2  —  2  c  v  cos.  5,  and  for  the  angle  v  A1  c  =  * ,  by 
which  it  deviates  from  the  circumference  of  the  wheel,  or  from  the 

tangent  Av,  we  have  sin.  *  =  c  8m'  S. 

The  water  ascends  on  the  float  with  a  retarded  velocity,  and  partakes 
of  the  velocity  of  rotation  v  of  the  wheel  at  the  same  time.  Having 
ascended  to  a  certain  height,  its  relative  velocity  is  lost,  and  it 
descends  with  an  accelerated  velocity,  so  that  at  last  it  arrives  at 
the  outer  extremity  Al  with  the  same  velocity  cl  with  which  it  com- 
menced its  ascent.  If  we  combine  the  relative  velocity  A1  cl  —  c, 
after  the  water  leaving  the  wheel  at  A,  with  the  velocity  of  the  cir- 
cumference Al  v  =  v  as  a  parallelogram  of  the  velocities,  we  have  in 
the  diagonal  A  w  =  w  the  absolute  velocity  of  the  water  leaving 
the  wheel.  This  velocity  is 

W  =  i/C2  -f  V2 2(7,  V  COS.  «, 

and,  therefore,  the  mechanical  effect,  retained  by  the  water,  and 
lost  for  useful  effect,  is 

Ll  =  £gQv 
If.  now,  we  deduct  this  loss  from  the  amount  of  effect  |-  Q  y  inhe- 


220  PONCELET'S  WHEELS. 

rent  in  the  water  before  its  entrance  on  the  wheel,  we  have  the  fol- 
lowing expression  for  the  theoretical  effect  of  the  wheel  : 


or,  as  c2  =  c*  +  v2  +  2c,  v  cos.  «,  .-.  L  =  2c^v  cos'  *  .  Qy,  or, 

__         _  ff  _ 
Cl  cos.  e  =  </c?  —  c2  sin.  82  =  >/c2  cos.  82  -f-  v2  —  2  c  v  cos.  8  =  c  cos. 
S  —  v,  and,  if  we  put  this  in  the  above  expression,  we  have  : 
r  _  2  v  (c  cos.  8  —  v)  ~  • 

g 
We  easily  perceive  that  the  effect  is  a  maximum  when  v  =  \  c 

cos.  3,  and  then  L  =  -  !  —  Qy.      Also,    the   loss   of  mechanical 

2g 
effect  is  null,  or  the  whole  mechanical  effect  available,  or  L  =  £-  Qy 

is  got  from  the  water  when  cos.  8  =  1,  or  when  8  =  0. 

Although  it  is  not  possible  to  make  the  angle  of  entrance  8=0, 
it  follows  from  this  that  8  should  not  be  a  large  angle  —  not  more 
than  30°,  if  a  good  effect  is  desired,  and  it  is  also  manifest  that 
the  velocity  of  rotation  of  the  wheel  should  be  only  a  little  less  than 
half  the  velocity  of  the  water  going  on  to  the  wheel,  that  the  effi- 
ciency may  be  the  greatest. 

§  122.  The  vertical  height  I/O,  to  which  the  water  ascends  on 

the  floats,  would  be  ^-  if  the  wheel  were  at  rest,  but  as  it  has  a 

2# 

velocity  of  rotation  v,  a  centrifugal  force  arises,  acting  nearly  in 
the  same  direction  as  gravity,  and  giving  rise  to  an  acceleration 

p,  which  may  be  represented  by  ^L,  if  ax  be  the  mean  radius  CM, 

a, 

and  vl  the  mean  velocity  of  the  wheel's  shrouding,  or  the  velocity 
of  the  point  M.     We  have  then  : 


and  hence,  the  height  of  ascent  in  question  Jil  = 


In  order  that  the  water  may  not  pass  over  the  top  at  0,  it  is  neces- 
sary that  the  shrouding  should  have  a  certain  depth  FO  =  d,  which 
is  determined  by  the  equation  d  =  LO  +  FL  =  h:  +  OF  —  OL 

=  Aj  +  a  —  a  cos.  ACF  =  -  £l  ---  \-  a  (1  —  cos.  x),  where  x  is 


the  angle  ACF  by  which  the  point  of  entrance  of  the  water  on 
the  wheel  deviates  from  the  lowest  point  of  the  wheel  F.  The 
thickness  of  the  stream  of  water  dl  is  to  be  added  to  this,  because 
the  particles  in  the  upper  stratum  must  rise  so  much  higher  than 


PONCELET'S  WHEELS.  221 

those  of  the  tower  stratum  on  the  assumption  of  a  mean  velocity. 
The  depth  of  the  shrouding  is,  therefore, 

c  2 
^  =  ^i  -\ — 5-  +  «  (1  —  cos.  x). 


The  width  of  the  wheel  is  equal  to  the  width  of  the  stream  of 

water;  or,  e  =  -JL-.     If  the  capacity  d  e  v.  of  the  wheel  be  made 
d^c 

\\  times  that  of  the  water  laid  on,  then  we  have  the  equation 
d  v1  =  §  di  c  to  2  dl  c,  and  hence  the  thickness  or  depth  of  the 

stream  laid  on  =  dl  =  J  —^  to  §  —  ^  .    Another  important  circum- 

c          '     c 

stance  in  reference  to  these  wheels  is  the  determination  of  the  points 
of  entrance  and  exit  of  the  water,  that  is  the  water  arc  AAV  which 
it  is  best  to  set  off  in  two  equal  portions  on  each  side  of  the  lowest 
point  of  the  wheel  F.  The  length  of  this  arc  depends  on  the  time 
necessary  for  the  ascent  and  descent  of  the  water  on  the  floats. 
To  find  this,  we  must  know  the  form  and  dimensions  of  the  floats. 
If  the  time  =  <,  then  we  may  put  AAt  =  2  x  a  =  v  £,  and  hence 
the  points  on  either  side  of  F,  at  which  the  water  enters  and  quits 

the  wheel,  are  at  a  distance  =  x  =  —  . 

§  123.  In  order  that  the  water,  when  it  has  reached  the  highest 
point  K,  Fig.  232,  may  not  run  over,  but 
fall  back  along  the  float,  the  inner  end  of  Fig.  232. 

the  float  -BTmust  not  overhang  the  float 
when  in  the  mean  position  FK;  but,  on 
the  other  hand,  that  the  float  may  not  be 
too  long,  the  end  K  of  the  float  must  not 
cut  the  inner  circumference  of  the  shroud- 
ing at  too  acute  an  angle.  Hence,  it  is 
best  to  give  the  inner  end  of  the  float  a 
vertical  position,  when  the  float  is  in  its 
mean  position.  Adopting  a  cylindrical 
form  of  float,  we  get  the  centre  of  the  cir- 
cular arc,  its  section,  by  drawing  MF  per- 
pendicular to  Fcv  and  OM  horizontal.  From  the  depth  of  shroud- 
ing FO  =  d,  we  have  the  radius  : 


COS.  e 

t  being  the  angle  MFO  =  cl  Fv. 

The  time  required  for  the  ascent  and  descent  of  the  water  on  the 
arc  FK  may  be  found  in  the  same  manner  as  the  time  of  oscillation 
of  a  pendulum,  by  substituting  for  the  accelerating  force  of  gravity 

the  sum  g  -{•  V-±-  of  this  acceleration,  and  that  of  centrifugal  force. 

This  time  may  be  found  exactly  by  the  method  given  Vol.  I.  §  246, 

19* 


222  PONCELET'S  WHEELS. 

by  putting  here,  as  there,  the  instant  of  time  required  to  move 
through  a  small  space : 


_  /!   .    Ml  +  cos.  »)\     \r_    _?_ 
V    "  Sr         }+]g'2n 


In  order  to  find  the  time  for  ascent  and  descent  in  the  arc  FK, 
we  have  to  substitute  for  *  the  central  angle  MGL,  which  may 
be  determined  from  the  angle  cl  Fv  =  FMS=  «,  and  the  radius 
MF=  MS=  r,  by  the  formula  : 
_       NG 

~ 


=  —  (2  cos.  t  —  1),  or  sin.  %  $  =  ^/cos.  t. 

We  have  now  the  time  tl  required  for  describing  the  whole  arc  FK, 
by  adding  together  all  the  values  of  the  expression  : 


when  for  cos.  <|>  we  substitute  in  succession  : 

COS.   t  €08.  ?*,  €08.  ?*  .  .  .  €08.  ^1       But 

n          n  n  n 

sin.  -  co8.  | 

cot.  t  +  cos.  5*  +  cos.  ?*  +  ..+  cos.^=  _  -  _  - 
n  n  n  n  $ 


*L*.,  and  hence  t,  =  [~|  +  A  (|  +    *    times  the  sum  of 


all 


cosines 

If  we  also  consider  that  the  whole  height  of  fall,  or  the  diameter 
MS  =  A,  that  MF  =  r,  and  that  g  +  V-L-  is  to  be  substituted  for  g 

the  force  of  gravity,  the  whole  time  for  the  rise  and  fall  of  the  water 
on  the  arc  FK  is 


and  the  length  of  the  water  arc  A Al  (Fig.  231),  is : 


§  122.  We  have  now  to  derive  rules  for  the  arrangement  and 
construction  of  Poncelet's  wheels  from  these  data.  We  can  only 
assume  the  height  of  fall  A,  the  quantity  of  water  Q,  and  the  num- 
ber of  revolutions  u  of  the  wheel,  as  given,  and  from  this  we  have  to 
deduce  the  velocity  of  rotation  v,  the  radius  of  the  wheel  a,  the  depth 


PONCELET'S  WHEELS.  223 

of  shrouding  d,  the  width  of  wheel  e,  and  the  angles  8,  t ,  x,  and  the 
velocity  <?x  of  the  water  at  the  beginning  of  its  ascent.  If  we  atten- 
tively consider  the  formulas  above  found,  we  perceive  that  they  do 
not  admit  of  a  direct  solution  of  the  problem,  but  that  the  method 
of  gradual  approximation  must  be  adopted. 

If  we  lay  on  the  water  in  a  horizontal  direction,  the  deviation  8  of 
the  direction  of  the  water-stream  from  the  periphery  of  the  wheel  is 
equal  to  the  distance  x  of  the  point  of  entrance  from  the  foot  of  the 
wheel.  In  the  first  place,  we  may  put,  as  an  approximation,  the 
velocity  of  the  water  entering  the  wheel  :<?  =  /«  >/% A,  and  from  this 
again,  the  velocity  of  rotation  of  the  wheel  v  =  \  c,  as  also  the  initial 
velocity  of  the  ascending  water  cl  ==  £  c,  we  have  hence  also  an  ap- 
proximate value  of  the  radius  a  =  — -,  and  the  same  for  the  depth 

nu 

of  shrouding  d  =  £L.  =  £  .  — ,  and,  hence,  also,  we  obtain  an  ap- 


proximate value  for  the  length  of  the  water  arc,  if  we  put  in  the  last 
formula  of  the  preceding  paragraph : 

*  =  *,  *+fm-*  =  0,  and  r  =  d,  then: 


and,  therefore, 


With  the  assistance  of  this  approximate  value  of  x  =  8,  the  cal- 
culations must  be  repeated,  using  the  more  exact  formulas,  and 
taking  for  the  depth  of  the  water-stream  d^  an  appropriate  value  of 
from  3  to  12  inches,  according  to  circumstances.  The  head  or 
pressure  is  then  only  h  —  dv  and  hence  the  velocity  of  the  water 
entering  the  wheel  is:  1.  c  =  ft  <S2g  (h  —  df,),  that  of  the  wheel. 
2.  v  =  £  c  cos.  S.  Again,  the  radius  of  the  wheel  3.  a  =  — - ;  for 

ft  It 

the  angle  «  made  by  the  circumference  of  the  wheel  with  the  end 
of  the  float, 

4.  cotff.  «  =  cotg.  « —  =  J  cotg.  S,  or  tang,  t  —  2  tang.  «; 

c  sin.  5 
and  the  initial  velocity  of  the  water  rising  on  the  float. 

5.  c,  =  c_^Ll=  _?_.  and  tf  instead  of  ^,  we  put  -,  the   depth 

sin.  i       cos.  i  al  a, 

of  shrouding, 

6.  d  =  d1  ^ fi — _ 1-  a  (1 — cos.  x) : 


224  EXPERIMENTS  WITH  PONCELET'S  WHEELS. 

and  hence  again  we  have  the  width  of  the  wheel: 

7.  e  =  -^-,  and  the  radius  of  the  curvature  of  the  floats: 

d  c 

8.  r  =         ,  and  the  angle  $: 

COS.  t 

9.  sin.  J  t  =  </cos. «,  and  lastly  the  length  of  the  water  arc, 
10. 


and  from  this  the  accurate  value  of: 
11,       /     i    *  +  *in-  *\  *u 

ax-(t+ — g— ;  go 


V 


•+; 


Even  after  these  values  have  been  found,  the  calculations  may  be 
repeated  on  the  more  accurate  foundations. 

Example.  It  is  required  to  ascertain  the  general  proportions  of  a  Poncelet  undershot 
wheel.  Gwen,  the  height  of  fall  4,5  feet,  the  quantity  of  water  40  cubic  feet  per  second. 
If  we  make  the  radius  a  =  '2h=  9  feet,  and  allow  the  thickness  of  the  stream  rf,  ^  £ 
h  =  0,75  feet,  and  further,  p  =  0,90,  then  the  velocity  of  discharge  c  =  0,9  ^/2g  (h  —  d,) 
=  0,9  X  8,02  ^3,75  =  7,218  X  1.936  X  J4  feet;  and,  therefore,  the  velocity  of  the 
wheel,  as  also  the  initial  velocity  of  the  water,  is  approximately  r=r,^$  c  =  7  feet. 
Hence  the  depth  of  shrouding  is,  nearly, 

d  =  $.  ^.4.  rf,  =  $.  3,04  +  0,75=  1,51  feet,  and  thearcx  =  »  =  ^l 


=  0,24,  and  the  angle  x°  corresponding  =  14°,  for  which,  however,  we  shall  take  15°. 
If  we  now  introduce  this  value  of  fr,  we  get,  more  accurately,  v  =  ^  c  cos.  >  =  7  cos.  15° 

=  6,762  feet,  and  hence,  the  number  of  revolutions  u  =  —  *  =  7,17.  It  follows,  there- 

ir  a 
fore,  that  tang.  «  =  2  tang.  J=2  .  0,26795  =  0,53590,  .-.  i  =  28°  11$',  and,  therefore, 

c,  =  _  67'62  _  =  7,67  feet    Again,  we  have  the  depth  of  shrouding  d  =  0,75  +  9 
cos.  28°,  11$- 

(I—cos.  15°)  J  --  Z^!  _  =  1,845   feet:    and   the   width    of    the    wheel 
~  2  (32,2  +  £  .  6,76s) 

3,80  feet     The  radius  of  curvature  of  the  floats  r 


0,75.14  _  cos.  28°,  11$' 

=  2,093  feet,  and  sin.  $  <j,=v/co*.  28°,  11$'  .-.  £  ,j,0  =  690,  51$',  and  .-.  <?°=  139°, 
43',  f  =2,4385,  sin.  <},  =  0,6466;  and,  lastly, 
X=  (2.4385+  2,4385  +0,6466X6,76_    /|093=  1^093 

V  8  /  A    18    *J  36,52  -s/  36,52 

=  0,2499,  and  x°=  14°,  197,  for  which  14$°  would  be  substituted  in  the  actual  con- 
struction of  the  wheel,  so  that  the  length  of  the  water  arc,  or  the  length  of  the  concen- 
tric curb  b  =  2  X  a  =  18  .  0,253  =  4$  feet,  or  2$  feet  on  each  side  of  the  lowest  point 
of  the  wheel. 

§  123.  Experiments  with  PonceleCs  Wheels.  —  Poncelet  himself 
instituted  experiments  on  the  useful  effect  of  his  water  wheels. 
These  are  minutely  detailed,  and  their  results  ascertained  in  his 
work  above  cited. 

The  first  experiments  were  made  with  a  model  wheel  of  20  inches 


RECENT  EXPERIMENTS.  225 

diameter.  It  was  of  wood,  had  20  floats,  about  T^  of  an  inch  thick, 
2£  inches  deep,  and  3  inches  wide.  The  greatest  effects  were  pro- 
duced when  the  velocity  of  the  wheel  =  0,5  that  of  the  water,  as 
indicated  by  theory,  and  then  the  efficiency  was  0,42  to  0,56,  the 
former  when  the  water  stream  was  kept  thin,  the  latter  when  this 
was  increased,  or  the  cells  of  the  wheel  hetter  filled.  Reckoning 
the  efficiency  by  the  height  due  to  the  velocity  of  the  water,  and  not 
by  the  actual  fall,  the  effect  rises  to  0,65  to  0,72. 

Poncelet  afterwards  experimented  on  a  water  wheel  erected  on  his 
principle,  measuring  the  effect  by  means  of  a  friction  brake,  and  the 
results  are  very  much  the  same  as  those  obtained  from  the  model. 
The  wheel  was  11  feet  in  diameter,  and  had  30  plate-iron  floats  of 
£  inch  thickness.  The  shroudings,  arms,  and  axle  of  the  wheel 
were  of  wood.  The  shrouding  was  14  inches  deep,  and  3  inches 
thick,  the  distance  between  them,  or  width  of  the  wheel,  28  inches. 
For  a  mean  head  of  4'  —  4",  and  8  inches  depth  of  water  stream, 
the  ratio  of  the  velocities  being  0,52,  the  efficiency  came  to  0,52, 
which  gives  0,60,  when  the  height  due  to  the  velocity,  instead  of 
the  total  fall,  is  made  the  basis  of  calculation.  Poncelet  makes  the 
following  deductions  from  his  series  of  experiments. 

The  best  velocity  ratio  -  is  0,55  ;*  but  this  may  vary  between 

0,50  and  0,60  without  material  diminution  of  the  useful  effect.  For 
falls  of  6'  —  6"  to  7'  —  6",  the  efficiency  n  =  0,5,  for  falls  of  5  feet 
to  6'  —  6",  the  efficiency  tj  =  0,55,  and  for  falls  of  less  than  5  feet 
,  =  0,60. 

The  useful  effect  may,  therefore,  be  represented,  in  the  first  case, 
by: 

Pv  =  0,96  (c  —  v)  v  Q  ft.  Ibs.,  in  the  second : 
Pv  =  1,06  (c  —  v)  v  Q  ft.  Ibs.,  and  in  the  third : 
Pv  =  1,15  (c  —  v)  v  Q  ft.  Ibs. 

Poncelet  gives  the  following  general  rules  for  the  construction 
and  arrangement  of  his  wheels,  deduced  from  his  experiments.  The 
distance  between  2  floats,  at  their  outer  extremity,  should  not  ex- 
ceed 8  to  10  inches,  and  the  radius  of  the  wheel  should  not  be  less 
than  3'  — 4"  (1  metre),  nor  more  than  8'  — 2"  (2|  metres).  The 
axis  of  the  water  stream  should  meet  the  periphery  of  the  wheel  at 
an  angle  of  24°  to  30°,  and  be  inclined  about  3°  to  the  horizon. 
The  offset  at  the  end  of  the  curb  should  be  sufficient  to  insure  the 
water's  free  escape  from  the  wheel,  and  the  space  left  between  the 
wheel  and  the  curb  would  not  exceed  f  inch. 

According  to  the  experiments,  the  efficiency  increases  with  the 
depth  of  the  water  stream  laid  on,  and,  therefore,  cseteris  paribus, 
as  the  filling  of  the  cells.  Further  experiments  prove  that  the  degree 
of  filling  of  the  cells  is  an  important  element  in  the  question. 

§  124.  Recent  Experiments. — Morin  has  quite  recently  instituted 

*  [This  is  the  same  ratio  as  that  found  by  the  Committee  of  the  Franklin  Institute  for 
the  velocity  of  an.  overshot  wheel  with  elbow  buckets. — AM.  ED.J 


226  RECENT  EXPERIMENTS. 

experiments  with  three  wooden  and  one  iron  wheel,  constructed  on 
Poncelet's  principle,  using  the  friction-brake.  They  were  made  with 
the  special  object  of  testing  the  advantages  of  a  curvilinear  course 
for  laying  on  the  water,  proposed  by  M.  Poncelet ;  as  also  for  the 
purpose  of  getting  more  exact  information  as  to  the  influence  of  the 
relative  dimensions  of  the  wheels,  for  in  several  wheels  that  have 
been  erected  according  to  Poncelet's  rule,  it  is  found  that,  when  the 
deviation  from  the  mean  velocity  is  considerable,  the  water  overruns 
the  floats.  (See  Comptes  Rendus,  1845,  t.  xxn.)  As  to  the  curved 
water  course,  its  object  was  to  lay  the  whole  of 

Fig.  233. the  water  on  to  the  wheel  without  impact,  and 

not  the  top  or  bottom  stratum  only.  When 
the  water  stream  is  straight  ABED,  Fig.  233, 
the  upper  layer  of  water  DE  meets  the  peri- 
phery of  the  wheel,  as  also  the  float,  at  a  dif- 
ferent angle  from  that  at  which  the  lower 
stratum  does ;  so  that  if  one  enters  without 
impact,  the  other  cannot  do  so.  If,  however, 
we  hollow  out  the  bottom  of  the  course  as  A  OB,  the  water  stream 
comes  upon  a  smaller  arc  BK,  and  the  difference  in  the  direction 
of  the  periphery,  of  the  wheel  and  the  layers  of  water  is  less,  and, 
therefore,  the  impact  is  less  than  when  the  water  stream  embraces 
the  arc  BE. 

The  three  wooden  wheels  were  respectively  5'  —  3",  8'  —  3",  and 
10'  —  3"  in  diameter.  The  diameter  of  the  iron  wheel  was  9'  —  3". 
The  buckets  were  of  sheet  iron.  The  first  three  wheels  were  16 
inches  wide,  and  the  other  was  32  inches.  The  depth  of  shrouding 
was  30  inches.  It  was  found  that  wooden  wheels,  having  very  little 
inertia,  moved  unsteadily,  and  hence  arose  a  loss  of  water.  The 
smallest  wheel  revolved  very  unsteadily,  and  for  a  fall  of  18  to  22 
inches,  the  cells  being  at  least  half  filled,  the  efficiency  was  0,485. 
Had  the  weight  of  the  wheel  been  greater,  its  efficiency  would  pro- 
bably have  been  0,55.  The  second  wheel,  having  a  fall  of  30 
inches,  gave  an  efficiency  =  0,60  to  0,62.  The  third  wheel  was 
used  to  make  experiments  on  different  lengths  of  floats.  It  appeared 
that  for  a  fall  of  22  inches,  a  length  of  17  inches,  and  for  a  fall  of 
28  inches,  a  length  of  24  inches,  is  too  little.  Poncelet's  curved  lead 
was  adapted  to  this  wheel,  and  it  was  found  that  the  efficiency  was 
increased,  and  also  that  the  degree  to  which  the  cells  are  filled,  might 
be  made  §  without  inconvenience. 

The  experiments  with  the  iron  wheel  were  instituted  with  falls 
of  4  feet  to  4J  feet,  and  of  3  feet,  the  wheel  being  free  from  back- 
water, and  with  a  fall  of  15  inches,  the  wheel  being  in  back-water. 
For  sluice-openings  of  6  inches,  8  inches,  10  inches,  and  11  inches, 
the  maximum  efficiency  was  0,52,  0,57,  0,60,  and  0,62  respect- 
ively, and  for  variations  in  the  number  of  revolutions  between  the 
limits  of  12  to  21,  13  to  21,  11  to  20,  and  12  to  19,  the  efficiency 
did  not  differ  more  than  T^,  T'?,  Jj,  and  £  from  the  maximum  values. 
From  the  results  of  these  experiments,  it  follows  that,  for  wheels 
with  the  hottow  water-lead,  the  effect  is  expressed  by  the  formula: 


SMALL  WHEELS. 


22T 


Fig.  234. 


Also,  that  the  best  velocity  ratio  -  =  0,5  to  0,55.     That  the  same 

effect  is  produced,  whether  the  water  in  the  race  be  5  inches  below, 
or  8  to  10  inches  above  the  bottom  of  the  wheel — that  the  efficiency 
falls  as  low  as  0,46,  if  the  wheel  be  in  back-water  to  the  depth  of 
half  the  depth  of  the  shrouding.  The  main  advantage  of  the  new 
form  of  lead  is,  that  the  wheel  may  vary  its  velocity  of  rotation 
within  wider  limits,  without  material  diminution  of  the  efficiency. 
Morin  considers  that,  for  falls  of  3  feet  to  4  feet,  a  breadth  of  shroud- 
ing equal  to  the  half  of  the  radius  is  a  good  proportion  to  adopt, 
and  that  the  capacity  of  the  wheel  should  be  double  that  corre- 
sponding to  the  water  to  be  laid  on,  i.  e.,  the  co-efficient  of  filling 

£  =  —~-  should  be  made  =  J.* 
dev 

Remark.  It  would  thus  appear  that  the  capacity  of  the  wheel  treated  in  our  last  ex- 
ample is  too  small,  and  that  it  would  have  been  better  to  have  made  rf,  =  0,5  feet,  and 
e  =  5,71  feet. 

§  125.  Small  Wheels.  —  Some  other  vertical  water  wheels  have 
been  applied,  besides  the  systems  we  have  now  discussed.  Very 
small  wheels  of  2  or  3  feet  diameter,  are  moved  by  the  pressure  or 
impact  of  water. 

D'Aubuisson  describes,  in  his  "Hydraulique,"  small  impact  wheels 
AGE,  Fig.  234,  with  falls  of  6  to  7 
metres,    often  to  be  met  with   in   the 
Pyrenees.     These  wheels  are  from  7 
to  10  feet  in  diameter,  and  have  24 
hollowed  floats.     Their  effect  is  about 
,73  of  that  of  an  overshot  wheel  of  the 
same  fall.     The  effect  of  such  a  wheel 
may  be  calculated  by  the   theory   of 
breast  wheels   above   given,  for   these 
wheels  are  nothing  more  than  breast 
wheels  with  a   great  impact  fall  and 
small  height,  during  which  the  water 
can  act  by  its  weight.     To  prevent  the 
spilling  of  the  water,  the  wheels  are  hung 
in  a  curb  with  close-fitting  sides.     Such 
wheels  may  be  very  neatly  made  of  iron, 
and    are   to  be   found  in  North   Wales. 
This  kind  of  wheel  is  very  commonly  em- 
ployed at  the  forges  in  the  Alps. 

Fig.  235  represents  a  wheel  erected  by 
Mr.  Mary,  and  described  in  the  "Tech-     !•! 

*  [The  Committee  of  the  Franklin  Institute  tried  curved,  oblique,  and  ellx>w  buckets 
successively  on  the  same  wheel*    They  found  the  ratio  of  effect  to  power  for  the  curved 
buckets  nearly  equal  to  that  for  elbow  buckets,  while  in  reference  to  the  vela 
wheel  they  are  much  inferior.     Elbow  buckets  gave  5.6,  curved  4.2,  and  oblique  3.7 
feet  per  second  velocity  of  wheel.— AM.  ED.] 


Fig.  235. 


228  HORIZONTAL  WATER  WHEELS. 

nologiste,  Sept.,  1845."  The  water  here  works  chiefly  by  pressure. 
Belanger  experimented  with  the  wheel,  and  reported  an  efficiency 
of  0,75  to  0,85  for  a  velocity  of  4  feet  per  second.  The  wheel  con- 
sists of  a  shrouding  of  plate  iron,  13  inches  wide  and  5  inches  deep, 
and  7'  —  6"  in  diameter,  and  having  six  elliptical  floats  strengthened 
by  ribs. 

The  curb  is  made  to  fit  very  accurately,  and  sheet  iron  fenders, 
fitting  close  to  the  wheel,  prevent  the  water  in  the  lead  from  escap- 
ing into  the  race.  The  power  with  which  such  a  wheel  revolves,  is, 
of  course,  the  product  of  the  weight  of  water,  measured  by  the  dif- 
ference of  level  in  the  lead  and  in  the  race,  by  the  area  of  the  float. 

Literature.  The  literature  treating  of  vertical  water  wheels  is  very  extensive;  but  there 
are  few  works  upon  the  subject  worthy  of  much  attention,  as  the  most  of  them  give 
very  superficial  and  even  erroneous  views  of  the  theory  of  these  wheels.  Eytelwein, 
in  his  "  Hydraulik,"  treats  very  generally  of  water  wheels.  Gerstner,  in  his  "Mechanik," 
treats  very  fully  of  undershot  wheels.  Langedorf 's  "  Hydraulik"  contains  little  on  this 
subject.  D'Aubuisson,  in  his  work  "  Hydraulique  a  1'usage  des  Ingenieurs,"  treats  very 
fully  of  overshot  wheels.  Navier  treats  water  wheels  in  detail  in  his  "Lemons,"  and  in 
his  edition  of  "  Belidor's  Architecture  Hydraulique."  In  Poncelet's  "  Cours  de  Mecanique 
appliquee,"  the  theory  of  water  wheels  is  briefly,  but  very  clearly,  set  forth.  In  the 
"Treatise  on  the  Manufactures  and  Machinery  of  Great  Britain,"  P.  Barlow  has  given 
details  on  the  construction  of  water  wheels,  but  has  not  entered  into  the  theory  of  their 
effects,  &c.  Very  complete  drawings  and  descriptions  of  good  wheels  are  given  in 
Armengaud's  "Trait6  pratique  de  Moteurs  hydrauliques  et  a  vapeur."  Nicholson's 
"  Practical  Mechanic,"  contains  some  useful  information  on  this  subject.  The  most  com- 
plete work  hitherto  published  on  vertical  water  wheels  is  Redtenbacher's  "Theorie  und 
Bau  der  Wasserrader,  Manheim,  1846."  Poncelet's  and  Morin's  Memoirs  have  been 
already  cited. 

[The  experiments  of  the  Franklin  Institute  are  contained  in  the  Journal  of  that  insti- 
tution for  1831-2  (vols.  7,  8,  &  9),  and  for  1841.  In  the  last-mentioned  volume,  the 
discussion  of  the  results  is  commenced,  but  has  not  yet  been  completed.  The  com- 
mittee, as  originally  constituted,  does  not  appear  to  have  given  its  attention  to  the  appli- 
cation of  mathematical  reasoning  to  the  observations  made  and  experiments  performed. 
Subsequent  European  experiments  have  consequently,  in  this  respect,  occupied  the  atten- 
tion of  physical  inquirers  to  the  exclusion  of  the  American. — AM.  ED.] 


L 


CHAPTER   V. 

OF   HORIZONTAL   WATER   WHEELS. 

§  126.  IN  horizontal  water  wheels,  the  water  produces  its  effect 
either  by  impact,  by  pressure,  or  by  reaction,  but  never  directly  by 
its  weight.  Hence,  horizontal  water  wheels  are  classified  as  impact 
wheels,  hydraulic  pressure  wheels,  and  reaction  wheels.  These 
wheels  are  now  very  commonly  designated  by  the  generic  term  tur- 
bines (Ger.  Kreiselrader). 

The  impact  wheels  have  plane  or  hollow  pallets,  on  which  the 
water  acts  more  or  less  perpendicularly.  The  pressure  wheels  have 
curved  buckets,  along  which  the  water  flows,  and  the  reaction  wheels 
have  as  their  type  a  close  pipe,  from  which  the  water  discharges 


IMPACT  WHEELS. 


229 


more  or  less  tangentially.  Pressure  wheels  and  reaction  wheels 
are  generally  very  similar  to  each  other  in  construction,  the  essen- 
tial difference  between  them  being,  that  in  the  former  the  cells  or 
conduits  between  two  adjacent  buckets  are  not  filled  up  by  the  water 
flowing  through  them,  while  in  reaction  wheels  the  section  is  quite 
filled. 

According  to  the  different  directions  in  which  the  water  moves  in 
the  conduits  of  pressure  and  reaction  wheels,  two  systems  arise. 
The  relative  motion  of  the  water  in  the  conduits  is  either  horizontal, 
or  in  a  plane  inclined  to  the  horizon,  and  usually  vertical. 

In  the  first  system,  there  are  to  be  distinguished  those  wheels  in 
which  the  water  flows  from  the  interior  to  the  exterior,  and  those  in 
which  the  water  takes  the  opposite  course ;  and  in  the  second  system, 
there  are  the  distinct  cases  of  the  water  flowing  from  above  down- 
wards, and  that  in  which  it  flows  from  below  upwards.  / 

Horizontal  water  wheels  in  which  the  water  flows  from  above 
downwards,  are  often  named  Dana/ides. 

§  127.  Impact  Wheels. — Impact  turbines,  as  shown  in  Fig.  236, 
are  the  simplest,  but  also  the  least  efficient  form  of  impact  wheels. 
They  consist  of  16  to  20  rectangular  floats  AB,  A^BV  &c.,  so  set 
upon  the  wheel  as  to  incline  50°  to  70°  to  the  horizon.  The  water 
is  laid  on  to  them  by  a  pyramidal  trough  EF,  inclined  from  40°  to 
20°,  so  that  the  water  impinges  nearly  at  right  angles  to  the  floats. 
Such  wheels  are  employed  for  falls  of  from  10  to  20  feet,  when  a 


Fig.  236. 


Fig.  237. 


great  number  of  revolutions  is  desired,  and  when  simplicity  of  con- 
struction is  a  greater  desideratum  than  efficiency.  Wheels  of  this 
form  are  met  with  in  all  mountainous  countries  of  Europe,  and  in 
the  north  of  Africa,  applied  as  mills  for  grinding  corn.  They  are 
made  from  3  to  5  feet  in  diameter,  the  buckets  being  about  15  inches 
deep,  and  8  to  10  inches  long. 

The  mechanical  effect  of  these  wheels  is  determined  according  to 
the  theory  of  the  impact  of  water,  as  follows.  The  velocity  Ac  =  c, 
Fig.  237,  of  the  water  impinging,  and  the  velocity  Av  =  v  of  the 
buckets  may  be  each  decomposed  into  two  velocities  expressed  by 
the  formulas 

VOL.  II.— 20 


230  IMPACT  WHEELS. 

c,  =  c  sin.  8,  c2  =  c  cos.  8,  vl  =  v  sin.  a,  and  v2  =  v  cos.  a, 
8  being  the  angle  c  AN,  by  which  the  direction  Ac  of  the  stream 
of  water  deviates  from  the  normal  AN,  and  o  the  angle  HAN  at 
which  the  normal  is  inclined  to  the  horizon,  or  by  which  the  direc- 
tion of  the  wheel's  motion  deviates  from  the  normal,  or  the  plane 
of  the  bucket  from  the  vertical.  The  component  velocity  cl  =  c 
sin.  8,  remains  unchanged,  as  its  direction  coincides  with  that  of 
the  plane  of  the  bucket;  the  component  c2  =  c  cos.  8,  is,  on  the  other 
hand,  changed  by  impact  into  v2  —  v  cos.  a,  as  the  bucket  moves 
away  in  the  direction  of  the  perpendicular  with  this  velocity.  The 
water,  therefore,  loses  by  impact  a  velocity 

c2  —  vz  =  c  cos.  8  —  v  cos.  o, 

and  the  corresponding  loss  of  effect  =  > —  —L  Q  y.    If, 

now,  we  deduct  from  the  whole  available  mechanical  effect  —  Q  y, 

the  above,  and  further,  the  effect, ' — '-^L  Q  y,  and 

sin.  82  +  v2  cos.  a2\  Q  ^  whicli  tte  water  flowing  away  with  the 

velocity  w  =  \/c2  sin.  82  +  v2  cos.  o2,  retains,  the  mechanical  effect 
communicated  by  the  wheel  is 

L  =  Pv  «  [c2—(c  cos.  8  —  v  cos.  a)2  —  (c2  sin.  82  +  v2 cos.  a2)]   0*. 

(c  COS.  8 V  COS.  a)  V  COS.  a       ^ 

*=  * '- Q  y. 

g 

To  get  the  maximum  effect,  we  must  make  cos.  8  =  1,  or  8  =  0, 
or  direct  the  stream  at  right  angles  to  the  bucket,  and  besides  this, 
as  in  other  similar  cases  already  treated,  we  must  make  v  cos.  a 

=  i  c.  or  v  =  -—?- The   maximum    effect    corresponding,   is 

2  COS.  a 

Pv  =  I  —  Q  y  a  i  h  Q  y,  or  the  half  of  the  entire  mechanical  effect 

available. 

§  128.  The  effect  of  impact  wheels  is  increased  by  surrounding 
the  buckets  with  a  projecting  border  or 
?• 238-  frame,  or  by  forming  them  like  spoons, 

as  shown  in  Fig.  238.  Vol.  I.  §  385 
explains  the  cause  of  this  increased 
effect,  but  we  may  here  determine  the 
amount  of  this  increase.  As  the  bucket 
moves  in  the  direction  of  the  stream 
with  the  velocity  v2  =  v  cos.  a,  the  rela- 
tive velocity  of  the  water  in  reference 
to  the  bucket  may  be  put: 

cl=  c  —  v2=  c  —  v  cos.  a, 
and  if  ft  =  the  angle  cl  0  c,  by  which 
the  water  is  turned  aside  from  its  ori- 


IMPACT  WHEELS.  231 

ginal  direction,  the  absolute  velocity  of  the  water  flowing  off: 
__  w  —  Vc?  +  t>a8  +  2  c1  vz  C08.J 

N/0  —  V  COS.  a)2  +  V2  COS.  az  +  2  (c  —  V  COS.  a)  V  COS.  a  COS.  0, 


and  hence,  the  corresponding  loss  of  effect: 

=  [c2  —  2  (c  —  v  cos.  o)  v 
he  effect  of  the  wheel  : 
Pv  =        ~    Qy  =  (1- 


=  [c2  —  2  (c  —  v  cos.  o)  v  cos.  a  (1  —  co«.  p)]  QL, 
and  the  effect  of  the  wheel  : 


x    ^y    '  g 

When  the  buckets  are  plane,  ft  =  90°,  .-.  cos.  ft  =  0,  and,  there- 
fore 

T  _  (c  —  V  COS.  a)  V  COS.  a.  ~ 

as  we  have  already  found,  though  by  an  entirely  different  method 
of  inquiry.     In  the  case  of  hollow  buckets,  ft  is  greater  than  90°, 
and,  therefore,  cos.  ft  is  negative,  and  hence  1  —  cos.  ft  is  greater 
than  1,  consequently  the  effect  is  greater  than 
in  plane  buckets.  Fi&  239- 

To  this  class  of  wheels  belong  those  termed 
in  France  rouets  volants,  upon  the  effect  of 
which  MM.  Piobert  and  Tardy  have  recorded 
experiments  in  a  work  entitled  "Experiences 
sur  les  Roues  hydrauliques  &  axe  vertical,  &c.9 
Paris,  1840."  The  following  are  results  of  ex- 
periments on  a  small  wheel  of  5  feet  diameter, 
8  inches  high,  having  20  curved  buckets,  Fig. 
239,  with  a  fall  of  14  feet  (measuring  from  surface  of  water  in  lead 
to  bottom  of  wheel),  and  with  10  cubic  feet  of  water  per  second: 

For  -  =  0,72,  7  =  0,16 ; 
"£-0,66,,  =  0,31; 

"^=0,56,^  =  0,40; 
c 

and  hence,  in  cases  in  which  the  velocity  ratio  -  does  not  much  differ 
from  0,6  :  Pv  =  0,75  (c  —  v  cos.  a)  v  C08'  *  Qy. 

t7 

Exatmle.  What  effect  may  be  expected  from  an  impact  turbine  with  hollow  buckets 
(Fig.  239 j.  there  being  6  cubic  feet  of  water,  and  a  fall  of  16  feet  at  disposition  ?  If  we 
neglect  the  depth  of  the  wheel  itself,  the  theoretical  velocity  of  entrance  of  the  water 
c  =  v/2gA  =  8,u;>  i/lG  =  32,08  feet,  and  if  the  inclination  of  the  trough  be  assumed  as 

20°,  the  most  advantageous  velocity  for  the  wheel  v  =  — f —  =  — - — -  =  I7    feet 

2  cos.  a,        cos.  20° 

and  hence,  from  the  above  formula,  the  effect  attainable  is 
T .=  P« g n  7S    c v cos-  *  — pl «"• "'  Qy  =  J  .  031  .  (512,09-15,96')  .6  .  62,5 «, 023 

X  257.4  .  375  =  2220  feet  Ibs. 


232  IMPACT  AND  REACTION  WHEELS. 

§  129.  Impact  and  Reaction  Wheels.  —  If  we  give  the  buckets 
greater  length,  and  form  them  to  such  a  hollow  curve,  that  the  water 
leaves  the  wheel  in  a  nearly  horizontal  direction,  the  water  then  not 
only  impinges  on  the  hucket,  but  exerts  a  pressure  on  it,  and,  there- 

fore, the  effect  of  the  wheel 

Fis-  24°-  is  greater  than  in  the  im- 

pact wheel.  The  theory  of 
such  wheels  is  merely  an 
extension  of  that  given  in 
§  127.  If  we  conceive  a 
normal  erected  at  the  point 
of  entrance  -4,  Fig.  240, 
and  if  we  again  put  the 
angle  c  AN  '=  8,  and  the 
angle  v  AN=  a,  we  have 
the  lost  velocity  arising 
from  impact  : 

c2  —  v2  =  c  cos.  6  —  v  cos.  o,  and  the  loss  of  effect  corresponding 
_  (c  cos.  8  —  cos.  a)2  * 

~2ff~       ~ 

The  velocity  with  which  the  water  begins  to  flow  down  the  buckets 
is  c1  +  c3  =  c  sin.  &  +  v  sin.  a,  and  if  we  put  the  height  BH,  through 
which  the  water  descends  on  the  bucket  =  hv  we  have  the  relative 
velocity  of  the  water  at  the  bottom  B  of  the  bucket  : 

c4  =  ^/(e1  +  c3)2  +  2ght  =  V(c  sin.  $  +  v  sin.  a)2  +  2ght. 
But  the  water  possesses  the  velocity  v  in  common  with  the  wheel, 
and,  therefore,  the  absolute  velocity  of  the  water  flowing  from  the 
wheel  :  w  =  -v/c,2  +  v2  —  2  c4  v  cos.  ©,  where  ©  =  the  angle  c4B  0,  at 
which  the  lowest  element  of  the  bucket  is  inclined  to  the  horizon. 
The  loss  of  effect  corresponding  to  this  is  : 
w*  Q 


If  we  deduct  these  two  losses  from  the  whole  available  effect,  we  get 
the  useful  effect  communicated  to  the  wheel  : 

L  =  Pv  =  [c2  —  (c  cos.  S  —  v  cos.  a)2  —  (c2  +  v*  —  2c4v  cos.  ®)]  -|p, 

in  which  we  have  to  substitute  for  c4  the  value  above  given. 

If  the  water  impinges  at  right  angles  8=0,  and 
c4  =  \/v2  sin.  a2  +  2ghv  and,  therefore, 

L  =  O2  —  (c  —  v  cos.  a)2  —  (c*  +  v*  —  2  c4v  cos.  ©)]  -|* 

=  [2  C  V  COS.  a  —  (1  +  COS.  a2)  VZ  -  V2  Sin.  a2  -  2^AX  +  2  V  COS.  ©  . 


& 


=  [(c  cos.  a  —  v)  v  —  ghj^  +  v  cos.  ©  \/v2  sin.  a2  +  2^rAj]  —  — 

In  order  that  the  water  may  produce  its  maximum  effect,  it  should 


PRESSURE  WHEELS.  233 

fall  dead  from  the  wheel,  or  w  should  =  0.     This  requires  that  0=0, 

and  c4  —  v,  i.  e.,  v2  sin.  a2  +  2#Aj  =  v2, .-.  v  =  ^'^9\ 

cos.  a 

§  130.  Pressure  Wheels.— If  the  water  is  to  be  laid  on  without 
impact,  then  v  cos.  a  must  =  c  cos.  8,  and  in  order  that  the  water 
may  quit  the  wheel,  deprived  of  its  vis  viva,  we  must  have :  ©  =  0, 
and  c4  =  v,  i.  e.  (c  sin.  8  +  v  sin.  a)2  +  2gh^  =  v2,  or  c2  sin.  82  +' 
2  c  v  sin.  a  sin.  8  +  2#A,  =  v2  cos.  a2  =  c2  cos.  82.  If,  again,  we  sub- 
tract from  both  sides :  2 \  c  v  cos.  a  cos.  8  =  2  c2  cos.  82,  then :  c2  sin.  b2 
—  2  c  v  (cos.  a  cos.  8  —  sin.  a  sin.  8)  +  2gJ^  =  —  c2  cos.  82,  or  c2  + 
2ghi  =  2  c  v  cos.  (a  -f  8),  and,  therefore, 

~~  2  c  cos.  (o  +  8)  ~~    c  cos.  $  ' 

in  which  A  is  the  velocity  of  the  water  at  entrance,  and,  therefore, 
A  +  At  the  whole  fall,  t  is  the  angle  c  A  v  between  the  direction  of 
the  water  and  that  of  the  wheel.     The  theoretical  effect  is,  in  the 
latter  case,  =  (h  +  AJ  Qy,  and  the 
efficiency  17  =  1,  because  there  is  Fig-  241. 

no  loss  from  any  cause.  When,  for 
such  a  wheel,  the  best  velocity  of 
rotation  v  has  been  found,  we  get 
the  requisite  position  of  the  buckets 
by  drawing  through  the  point  of 
entrance  A,  Fig.  241,  a  line  paral- 
lel to  v  c,  completing  the  parallelo- 
gram A  v  c  cr  The  side  A  cv  thus 
given,  gives  the  relative  velocity  cv 
with  which  water  begins  to  descend  along  the  bucket,  in  magnitude 
and  direction,  and  also  the  direction  of  the  upper  element  of  the 
bucket. 

That  the  water  may  flow  unimpeded  through  the  openings  BBV 
&c.,  the  foot  of  the  buckets  must  have  a  slight  inclination  to  the 
horizon.  If  we  put  the  mean  radius  of  the  wheel  =  a,  and  the  mean 
length  of  the  buckets,  measured  on  the  radius,  =  I,  we  may  put  the 
section  of  the  orifice  of  discharge  =  BN .  I  =  BBl  sin.  0  .  /,  and, 
therefore,  the  section  of  the  united  orifices  of  the  wheel  =  2  *  a  I 
sin.  0.  If,  again,  cz  =  the  relative  velocity  with  which  the  water 
arrives  at  the  bottom  of  the  wheel,  or,  if 

c2  =  -v/02  +  v2  —  2  c  v  cos.  $  +  2^7  A1? 
we  have  2  A  a  I  sin.  &  =  -*-,  and,  therefore,  for  the  requisite  angle 

sin.  &  =  0        ,    . 

2  *  a  I  c2 

Remark.  According  to  the  theory  of  the  impact  of  water,  or  of  hydraulic  presture,  ex- 
pounded in  our  first  volume,  it  is  not  necessary  that  v  cot.  a  =  c  cos.  $,  or,  which  amounts 
to  the  same,  that  the  component  c,  of  the  velocity,  should  fall  in  the  direction  of  the 
bucket.  According  to  Vol.  I.  §  43,  the  relative  velocity  c,  of  the  water  in  reference  to 
the  bucket  jlB,  Fig.  242,  is  the  diagonal  of  the  parallelogram  constructed  from  the  abso- 

20* 


234  BORDA'S  TURBINE. 

lute  velocity  of  the  water  c,  and  the  velocity  of  the  wheel  «,  taken  in  the  opposite  direc- 
tion; therefore,  c,  =  v/ca+l>3  —  2cvcos.<p.  If,  now,  the  direction,  but  not  the  magnitude, 
of  this  velocity  be  changed  by  the  shock  on  the  bucket,  we  have  the  relative  velocity 
at  discharge,  after  descent  through  the  height 


BH  =  A,,  c2  =  v/cT+'SgA,  =  v^-ftr1  —  2cvcos.  <p+2ght  :  lastly,  that  the  whole  effect 
may  be  taken  up  from  the  water,  we  have  to  make  : 

r2  =  v,  or  c2  -j-  if  —  2  c  v  cos.  $  -f-  2gh  ,  =  t;2,  therefore, 


_ 
2ccos.<j>  c  cos.  <}> 

§  131.  Bordas  Turbine.  —  The  wheels  discussed  in  the  last  para- 
graph, are  called  Bordas  turbines,  from  their  having  been  the  sug- 
gestion of  that  distinguished  officer  and  philosopher.  Their  con- 

Fig.  242.  Fig.  243. 


struction  is  shown  by  Fig.  243,  which  is  a  sketch  of  one  driving  6 
amalgamation  barrels,  at  the  silver  mines  of  Huelgoat  in  Brittany. 
The  curved  buckets  are  composed  of  three  beech  boards  put  care- 
fully together,  and  the  inner  and  outer  casings  are  composed  of 
staves,  the  outer  one  being  bound  by  two  iron  hoops.  The  diameter 
of  the  wheel  is  5  feet.  The  buckets  14  inches  long  or  deep,  and 
16 J-  inches  wide.  There  are  20  of  them.  The  fall  was  16'  —  3", 
and  the  wheel  makes  40  revolutions  per  minute. 

There  are  no  good  experiments  on  the  efficiency  of  Borda's  tur- 
bines. Borda  gives  0,75  of  the  theoretical  effect  as  the  useful 
effect,  or  L  =  0,75  .  [A  +  ^  —  (c  cos.  S _  —  v  cos.  a)2  —  ^02]  Q  y. 
Poncelet  very  justly  remarks  that  it  is  advisable  to  make  the  diame- 
ter and  the  height  of  the  wheels  as  great  as  possible,  so  as  to  curtail 
the  length  of  bucket,  that  is,  bringing  the  outer  and  inner  casings 
near  to  each  other.  By  giving  height  to  the  wheel,  the  fall  due  to 
the  velocity  is  diminished,  and,  therefore,  the  velocity  of  the  water 
and  of  the  wheel  is  less.  By  keeping  the  diameter  great,  the 
number  of  revolutions  falls  out  less,  and  as  for  a  larger  wheel,  the 
capacity  remaining  the  same,  the  width  of  the  wheel  may  be  less, 
and  then  the  difference  of  velocity  of  the  particles  of  water  adjacent 
to  each  other  will  be  less. 

Example.  What  quantity  of  water  must  be  supplied  to  a  Borda's  turbine,  constructed 
as  shown  in  Fig.  243,  which,  with  a  fall  of  15  feet,  is  to  drive  a  pair  of  millstones  re- 
quiring 2  horse  power?  Suppose  the  wheel  to  be  Ij  feet  high,  then  the  theoretical 
velocity  of  entrance  of  the  water : 

— 1,75  =  8,02  v/13,25  =29,19  feet. 


ROUES  EN  CUVES.  235 

If  the  water  be  laid  on  at  an  angle  of  30°  to  the  horizon,  then  the  least  velocity  of  rota* 

ti0n  ^  "=  *-T£¥  =  29^^=  19'L     If  the  —  •»"»  without  shock, 
the  velocity  with  which  it  begins  its  descent  along  the  bucket  is  : 


*  —  2  c  »  "••  ?  =  i/(a+v'  —  <a  —  2ghl  =  ^/v>—2ghi  =  1  5,88  feet.  For  the 
angle  ^,  at  which  the  upper  element  of  the  buckets  must  incline  to  the  horizon,  we  have  : 

!^i  =  1.-.  sin.  4  =  £M?  sin.  30°  =  0,9189  .-.  ^  =  66°,  46'.     If  we  give  the  bottom  of 

««.<}>         C,  10,88 

the  bucket  an  inclination  of  25°  to  the  horizon,  we  get  for  the  absolute  velocity  of  the 
water  flowing  away  : 

w  =  2  r  sin.  £  =  2.19,1  tin.  12£°  =  8,2  ft, 
and,  hence,  the  effect  : 

L  =  }  (h+  *.—  ^)  Q>"=i  (l5  -|!T).  62,5  Q  =  654  Q. 

That  we  may  have  2  horse  power,  or  1  100  feet  pounds  per  second,  we  must  have 

=  1,7  cubic  feet  of  water  per  second.  If  the  mean  radius  (measured  to  the  centre  of 
the  buckets)  of  the  wheel  be  2  feet,  and  if  the  water  space  be  6  inches  wide,  we  get 
the  united  areas  of  section  of  the  orifices  of  discharge  at  the  bottom  of  the  wheel 
=  2  it  a  I  sin.  e  =ic  .  4  .  £  sin.  25°  =  2,65  square  feet,  which  is  quite  sufficient  to  pass 
1,7  cubic  feet  of  water  per  second,  with  a  velocity  of  19  feet. 

§  132.  Roues  en  Cuves.  —  To  this  category  of  turbines  belong 
those  horizontal  wheels  enclosed  in  a  pit  or  well,  frequently  met 
with  in  the  south  of  France,  and  called  roues  en  cuves  (Ger.  Kufen- 
rader).  They  are  described  by  Belidor  in  the  "  Architecture  Hy- 
draulique,"  by  D'Aubuisson  in  his  "  Hydraulique,"  and  Piobert  and 
Tardy,  in  the  work  already  cited,  have  given  the  results  of  experi- 
ments instituted  on  one  of  these  wheels.  These  wheels  are  very 
similar  in  form  to  those  last  described  (Fig.  239).  They  are  gene- 
rally 1  metre  in  diameter,  and  have  9  curved  buckets.  They  are 
made  of  only  two  pieces,  and  are  bound  together  by  iron  hoops. 
The  axis  CD  (Fig.  244)  stands  on  a  pivot, 
the  footstep  of  which  is  on  a  lever  CO,  by  Fig-  244. 

which  the  wheel  may  be  raised  and  lowered 
as  the  millstone  may  require.  The  wheel  is 
near  the  bottom  of  a  well,  2  metres  deep,  and 
1,02  metres  in  diameter.  The  water  comes 
into  the  well  by  a  lead  laid  tangentially  to  it, 
about  13  feet  long,  the  breadth  at  the  outer 
extremity  being  2'  —  6",  and  at  the  entrance 
to  the  well  about  10  inches.  The  water  flows 
in  with  a  great  velocity,  acquires  a  rotary 
motion  in  the  wheel  chamber,  and  acts  by 
impact  and  pressure  on  the  wheel  buckets, 
flowing  through  it  into  the  tail-race.  There  is  evidently  a  great  loss 
of  water  in  such  wheels,  and  their  efficiency  is  consequently  small. 
Piobert  and  Tardy  found  an  efficiency  of  0,27  for  a  well  wheel  at 
Toulouse,  the  fall  being  10  feet,  with  13J  cubic  feet  of  water  per 
second,  and  the  number  of  revolutions  u  =  100.  For  u  =  120,  the 
efficiency  n  was  =  0,22,  and  for  u  =  133,  ?  =  0,15.  The  wheels  of 
the  Basacle  mill,  at  Toulouse,  give  an  efficiency  of  0,18. 


236       BURDIN'S  TURBINES — EFFECT  OF  CENTRIFUGAL  FORCE;. 

D'Aubuisson  mentions  that  wheels  of  this  kind  have  been  erected 
recently,  the  wheel  being  put  immediately  under  the  bottom,  and 
made  of  somewhat  greater  diameter  than  the  well.  The  pyramidal 
trough  for  laying  on  the  water  is  much  shortened,  and  by  these 
means  the  efficiency  has  been  raised  to  0,25.  These  wheels  are, 
therefore,  at  best,  inferior  to  the  impact  wheels  already  treated  of. 

§  133.  Burdin's  Turbines. — M.  Burdin,  a  French  engineer  of 
mines,  proposed  what  he  terms  a  "  turbine  &  Evacuation  alternative." 
They  are  the  best  wheels  of  the  category  now  under  examination. 
They  differ  from  Borda's  wheels  only  in  this  essential,  namely,  that 
the  water  enters  them  at  various  points  simultaneously,  and  that  the 
orifices  of  discharge  are  distributed  over  3  concentric  rings.  This 
latter  arrangement  is  adopted,  that  the  water,  discharged  with  a 
small  absolute  velocity,  may  not  hinder  the  revolution  of  the  wheel. 

The  first  wheel  of  this  kind 
was  erected  by  Burdin  for  a 
mill  at  Pont-Gibaud,  and  is 
described  in  the  "Annales  des 
Mines,  in.  serie,  t.  in."  Fig. 
245  represents  a  plan  of  this 
wheel.  ABD  is  the  pen- 
trough  immediately  above  the 
wheel,  having  a  series  of  ori- 
fices EF'm  the  bottom,  through 
which  the  water  is  laid  on  to 
the  wheel  with  a  slight  incli- 
nation. The  wheel  revolving  on  the  axis  c  consists  of  a  series  of 
conduits,  the  entrances  to  which  make  together  the  annular  space 
GrBH,  which  moves  accurately  under  the  arc  EF  formed  by  the 
trough-openings,  so  that  the  water  passes  unimpeded  from  the  one 
into  the  other.  The  conduits  (Fr.  couloirs]  are  vertical  at  the  upper 
end,  and  nearly  horizontal,  and  tangential  at  the  bottom.  The  lower 
ends  are  brought  into  three  distinct  rings,  so  that  the  third  of  the 
number  of  entrances  only  discharge  in  the  ring  vertically  under 
them ;  one-third,  as  K,  discharge  within,  the  others,  as  L,  discharge 
outside  this  ring. 

From  the  experiments  made  on  the  turbine  erected  at  Pont-Gibaud 
by  Burdin,  it  appears  that  for  3  cubic  feet  of  water  per  second,  and 
a  fall  of  10,35  feet,  the  efficiency  was  0,67.  The  impact  turbine 
formerly  in  the  same  position  consumed  3  times  this  quantity  of 
water  to  produce  the  same  effect.  The  diameter  of  the  wheel  was 
4,6  feet,  and  the  depth  15  inches.  The  number  of  buckets  36. 

§  134.  Effect  of  Centrifugal  Force. — In  the  turbines  hitherto  under 
consideration,  the  water  moves  nearly,  if  not  exactly,  on  a  cylindri- 
cal surface,  and,  therefore,  each  element  of  water  retains  the  same 
relative  position  to  the  axis,  or  at  least  does  not  vary  it  much.  But 
we  have  now  to  consider  wheels,  in  which  the  water,  besides  a  ro- 
tary and  vertical  motion,  possesses  a  motion  inwards  or  outwards  in 
reference  to  the  axis,  and  more  or  less  radial.  The  peculiarity  of 


« 

PONCELET'S  TURBINE.  237 

such  turbines  is,  that  their  motion  depends  on  the  centrifugal  force 
of  the  water,  so  that  they  might  be  termed  centrifugal  turbines. 
Before  entering  on  a  discussion  of  these  wheels,  it  will  be  well  to 
investigate  the  effect  of  the  water's  centrifugal  force,  when  its  motion 
is  in  a  spiral  line  round  a  centre,  or  when  the  motion  is  radial  and 
rotary  at  the  same  time.  The  centrifugal  force  of  a  body  of  the 
weight  Or,  revolving  at  a  distance  y,  with  an  angular  velocity  «, 

round  a  given  point,  is  F  = ^  (Vol.  I.  §  231).      If  this  weight 

moves  also  a  small  distance  a  radially  outwards,  or  inwards,  then 
this  force  will  have  produced,  or  absorbed,  an  amount  of  mechanical 

effect  represented  by :  F  a  = ^—  .     If,  then,  we  assume  that  the 

g 

motion  commences  in  the  centre  of  rotation,  and  continues  radially 
outwards,  so  that  ultimately  the  distance  of  the  weight  from  the  axis 
=  r,  we  may  ascertain  the  mechanical  effect  produced  by  the  cen- 
trifugal force  by  substituting  in  the  last  formula  a  =  -  y,  introducing 

n 

successively, however,-, — ,  —  ...  — ,  and  uniting  the  mechanical 

n    n    n  n 

effects  resulting  by  summation.     Hence  the  mechanical  effect  in 
^ _.  question  is : 

>  -r  to2  G 


ng 


.. 
n  g  n  g  z 

or,  as  we  must  assume  n  infinite: 

L=—^    ^!  =  ^2     a=  —  G- 
n*g    '  2  "   2ff  '  2g 

when  v  is  the  velocity  of  rotation  «  r  of  the  body  at  the  extreme 
point  of  its  motion.  As  this  mechanical  effect  is  produced  by  the 
centrifugal  force  when  the  motion  is  from  within  outivards,  it  must 
be  consumed  when  the  motion  is  from  without  inwards.  If  the 
body  does  not  come  to  the  centre  at  the  end  of  its  motion,  but 
remains  at  a  distance  rl  from  it,  then  there  remains  an  amount  of 

effect  iJ-i-  0-t  and  the  body  consumes,  therefore,  only  the  effect 

i  =  ^_^  2)^ 


if  vl  represent  the  velocity  of  rotation  at  the  distance  rl  or  end  of 
the  motion,  as  v  represents  it  at  the  distance  r  or  commencement  of 
the  motion.  If  the  motion  is  from  within  outwards,  then  the  effect 

produced  by  centrifugal  force  is  L  ={  —  -  —  -  \  Cr. 

§  135.  Poncelet's  Turbine.—  One  of  the  most  simple  horizontal 
wheels,  in  which  centrifugal  force  influences  the  working,  is  Ponce- 


238 


PONCELET'S  TURBINE. 


let's  turbine,  shown  in  Fig.  246,  in  plan.  This  turbine  has  curved 
buckets  between  shroudings,  and  is, 
in  fact,  one  of  Poncelet's  undershot 
wheels,  laid  on  its  side.  The  water 
is  laid  on  by  a  trough  AD  nearly  tan- 
gentially,  and  runs  along  .the  curved 
bucket  to  discharge  itself  in  the  inte- 
rior. That  the  effect  of  the  water  on 
the  wheel  may  be  a  maximum,  it  is 
necessary  that  the  water  should  enter 
without  shock  and  discharge  into  the 
interior  deprived  of  its  vis  viva.  The 
direction  of  the  end  of  the  bucket  J., 
insuring  no  shock,  is  determined  ex- 
actly as  for  Poncelet's  undershot  wheel, 
by  constructing  a  triangle  with  the  velocity  v  of  the  wheel  and  that 
c  of  the  water  entering  and  drawing  Acl  parallel  to  the  side  vc. 
The  relative  velocity  Acl  with  which  the  water  enters  the  wheel  is : 
cl  =  v/e2  +  v2  —  2  c  v  cos.  8,  8  being  the  angle  c  A  v  by  which  the 
direction  of  the  stream  of  water  deviates  from  the  tangent  to 
the  circumference  of  the  wheel.  This  velocity  is,  however,  dimi- 
nished by  centrifugal  force  during  the  motion  of  the  water  on  the 
bucket,  and,  therefore,  the  relative  velocity  Bc3  =  c3  with  which 
the*  water  comes  to  the  inside  of  the  wheel,  is  less  than  the  above 
velocity  cr  According  to  the  result  of  the  investigation  in  the 
last  paragraph,  the  water  loses  an  amount  of  effect  represented  by 

(v  ~~  ri  \  Qy,  or  — ~Vl  in  pressure  or  velocity  height,  v  being  the 
2^jr     /  2g 

velocity  of  rotation  at  the  commencement,  and  v,  that  at  end  of  the 
motion.     If,  therefore,  ^-  be  the  height  due  to  the  velocity  at  the 

c  a 
entrance  A,  and  ^-  that  at  the  exit  J5,  we  have 

;,s,  or  as 


•£-  =  -x ( « TT-  1 5  and>  therefore,  c*  «=  c*  — 

2g      2<7       \2<7      %g' 

c  2  =  e3  4-  v8  —  2  c  v  cos.  < 


?o«.  8,  <?2a  =  e*  +  vf  —  2  c  v  cos.  8,  and 
c2  =  </c*  +  v,a  —  2  c  v  cos.  8,  it  being  constantly  borne  in  mind  that 
r  is  the  velocity  of  rotation  at  the  outer  periphery,  and  i\  that  at 
the  inner.  In  order  to  rob  the  water  of  all  its  vis  viva,  the  end  B 
of  the  bucket  should  be  laid  tangentially  to  the  inner  periphery  of 
the  wheel,  and  also  c3  should  be  made 

=  t'j,  or  e8  -f-  t>i2  —  2  c  v  cos.  8  =  t'j2,  i.  e.,  v  cos.  $  =»  -. 

For  the  sake  of  an  unimpeded  discharge  of  the  water  to  the  inte- 
rior, the  angle  8t  at  which  the  inner  end  of  the  bucket  cuts  the  wheel, 
must  be  made  15°  to  30°,  and,  hence,  the  absolute  velocity  of  the 
water  discharged  w  =  \/c*  +  v  *  —  2  c3  vt  cos.  «„  or,  if  we  assume 


DANA1DES.  239 

v  cos.  3  =  |,  or,  vl  =  <?„  w  =  2  v,  »zw.  -|i}  and  the  loss  of  mechanical 


effect  corresponding  is  : 

I' 

lastly,  the  remaining  useful  effect  of  the  wheel  : 


According  to  Poncelet,  these  wheels  should  give  an  efficiency  of 
0,65  to  0,75. 

§  136.  Danatdes.  —  We  shall  next  treat  of  horizontal  wheels  which 
have  more  or  less  the  form  of  an  inverted  cone,  and  which  are 
termed  in  France  roues  a  poires,  or  Danaldes.     Belidor  describes 
them   in   the    "Architecture   hydraulique." 
Fig.  247  represents   the   general   arrange-  Fig.  247. 

ment  of  these  wheels.  They  consist  essen- 
tially of  a  vertical  axis,  with  a  double  conical 
casing  attached.  The  space  between  the 
casing  is  intersected  by  division  plates,  form- 


ing conduits  running  from  top  to  bottom. 
The  water  is  laid  on  by  a  trough  A  at  top, 
and  flows  off  through  the  bottom  of  the  cone 


.at  JE,  near  to  the  axis,  after  having  passed 
through  the  conduits  above  mentioned.  In 
the  simplest  form  of  wheels,  the  division 
plates  are  plane  surfaces,  running  vertically  ; 
in  other  cases  they  are  spiral  or  screw- 
formed.  Belidor  describes  the  wheel  without 
the  outer  casing,  but  the  wheel  is  placed  in 

a  conical  vessel  fitting  pretty  closely  to  the  blades,  or  division  plates. 
In  these  wheels,  gravity  and  centrifugal  force  act  simultaneously 
on  the  water.  If  the  water  enters  the  wheel  with  the  relative  velo- 
city Cj  above,  at  the  point  jB,  the  velocity  of  rotation  of  which  is  v, 
and  flows,  in  the  wheel,  through  a  height  hv  the  velocity  at  the 
bottom  of  the  wheel  near  the  axis  will  be  <?2,  determined  by  the  for- 
mula c2  =  c2  +  2ghl  —  v2.  In  order  that  this  may  be  0,  we  must 
have  v2  =  c2  +  2ghr  Further,  that  the  water  may  enter  the  wheel 
without  shock,  the  horizontal  component  of  its  velocity  must  equal 
the  velocity  of  rotation,  that  is,  c  cos.  8  =  v,  8  being  the  inclination 
of  the  stream  of  water  to  the  horizon. 

The  relative  velocity  of  entrance  is  cl  =  c  sin.  8,  and,  therefore, 
the  above  equation  of  condition  becomes, 

c2  cos.  82  =  c2  sin.  82-f  2ghv  i.  e.,  c2  cos.  28=  2ghv 

The  fall  necessary  for  the  velocity  is,  therefore,  A2  =  —  = ^-- 

£Q       cos»  £  o 

If,  now,  the  whole  fall  h,  +      \     =  h,  then  the  depth  of  the 
cos.  -  6 


240  REACTION  OF  WATER. 

wheel  A.  =    ^  cos'  2  8-,  and  the  height  due  to  the  velocity: 
1  +  cos.  2  s 

2~ 


By  this  arrangement  there  is  no  loss  of  mechanical  effect,  but  as 
the  axis  lias  a  certain  sectional  area,  and  the  water  too  requires  a 
certain  area  of  orifice  for  discharge,  and  thus  the  water  can  only  be 
brought  to  within  a  certain  distance  of  the  axis,  its  vis  viva  cannot 
be  entirely  taken  up,  so  that  the  efficiency  is  not  nearly  1,0. 

Remark.  The  wheel  just  described  is  known  as  Burdin's  Danai'de.  The  older  Danai'de 
of  Manouri  d'Ectot  was  differently  constructed,  though  in  principle  it  was  the  same. 
This  wheel  consisted  of  a  sheet  iron  cylinder,  with  an  orifice  in  the  bottom  for  the  dis- 
charge of  the  water,  and  through  which  the  axis  passed.  In  this  hollow  cylinder,  there 
is  placed  a  closed  cylinder  in  such  a  position  as  to  leave  an  annular  space  between  it 
and  that  first  mentioned,  and  also  a  space  between  the  bottoms  of  the  two.  This  latter 
is  divided  by  plates  and  buckets  placed  vertically  and  radially  into  a  series  of  compart- 
ments. The  water  is  laid  on  tangentially  into  the  space  between  the  two  cylinders, 
descends  along  the  surface  to  the  bottom,  inducing  a  rotary  motion  of  the  whole  appa- 
ratus. In  this  manner  it  flowed  gradually  to  the  bottom,  and  from  thence  reached  the 
orifice  of  discharge.  See  "  Dictionnaire  des  Sciences  mathematiques  par  Montferrier, 
art.  Danai'de/' 

This  form  of  Danai'de  has  -been  recently  perfected  by  Mr.  James  Thomson,  of  Glas- 
gow, so  that  the  efficiency  of  a  model  has  been  proved  to  be  0,85. 

§  137.  Reactiofi  of  Water.  —  Before  proceeding  with  the  descrip- 
tion and  investigation  of  the  theory  of  reaction  wheels,  it  is  neces- 
sary that  we  should  illustrate  the  nature  of  the  reaction  of  water  in 
its  discharge  from  vessels.     As  a  solid  body  endowed  with  an  acce- 
lerated motion,  it  reacts  in  the  opposite  direction  with  a  force  equal 
to  the  moving  force,  so  it  is  in  the  case  of  water  when  it  issues  from 
a  vessel  with  an  accelerated  motion  from  the  orifice.     This  accelera- 
tion always  takes  place  when  the  area  of  the 
Fig.  248.  orifice  is  less  than  the  area  of  the  vessel,  or 

the  velocity  of  discharge  greater  than  the  velo- 
city of  the  water  through  the  vessel.  On  these 
grounds  the  vertical  pressure  of  the  water  in 
the  vessel  ERF,  Fig.  248,  from  which  the 
water  flows  downwards  at  F,  is  less  than  the 
weight  of  the  mass  of  water  in  the  vessel.  The 
decrease  of  this  force,  or  the  reaction  of  the 
water  flowing  away,  may  be  determined  as  fol- 
lows. If  the  horizontal  layer  AB  of  the  water 
flowing  out,  has  a  variable  section  Q-,  a  varia- 
ble thickness  z,  and  a  variable  acceleration  p,  its  weight  is  Gr  x  y 

and  its  mass  =  —  —  ,  and,  therefore,  its  reaction  K=  —  ?_?  .  p.    If, 

9  9 

now,  w  represent  the  variable  velocity  of  the  layer  of  water,  and  * 
its  increase  in  passing  through  the  elementary  distance  x,  we  have 

(according  to  Vol.  I.  §  19)  p  x  =  Wx,  and,  therefore,  K  =  —  -  wx. 

9 


REACTION  OF  WATER. 


241 


If  F  be  the  area  of  the  orifice,  and  v  the  velocity  of  discharge,  then 

G-w  =»  Fv,  and,  therefore,  K=  —?-vx.     To  obtain  the  reaction  of 

9 

all  the  layers  of  water,  we  must  substitute  in  the  last  expression  for 
x,  the  increments  of  velocity  *v  *2,  *3,  .  .  .  Xn  of  all  the  layers  of 
water,  and  sum  the  results.  The  reaction  of  the  whole  mass  is  thus 

P=  —  y  (*x  +  *2  -f  .  .  .  +  *n  )•     If  c  be  the  velocity  of  entrance 

of  the  water,  the  sum  of  all  the  increments  of  velocity  =  v  —  c, 
and,  therefore,  the  reaction  required: 

(v-c) 


.9  9 

Q  being  the  quantity  discharged  per  second.      If,  however,  the 

orifice  F  be  very  small  compared  with  the  surface  HR,  then  c  may 
be  neglected,  compared  with  v,  and 

P  =  v-  FY  =  2  .  J!_  F7  =  2A  .  Fy. 

So  that  the  reaction  is  as  great  as  the  vertical  impact  of  the  water 
on  a  plane  surface  (Vol.  I.  §  385),  that  is,  equal  to  the  weight  of  a 
column  of  water,  the  basis  of  which  is  the  area  F  of  the  orifice,  or 
of  the  stream,  and  whose  height  is  double  the  height  due  to  the  velo- 
city (2  h)  of  the  water  discharged. 

If  the  water  flow  out  by  the  side  of  the  vessel,  as  shown  in  Fig. 
249,  the  direction  of  the  reaction  is  then  horizontal,  and  the  amount 

is  in  like  manner  =  —  Fy.     If  the  water  vein  be  contracted,  and  if 

9 
a  be  the  co-efficient  of  contraction,  then,  instead  of  F,  we  must  put 

Fa,  or  P  =  —  .  a  Fy. 
9 

Remark.  Mr.  Peter  Ewart,  of  Manchester,  made 


experiments  to  test  this  result,  ((i  Memoirs  of  the 
Manchester  Phil.  Soc.,"  Vol.  II.)  The  vessel 
HRF  (Fig.  249)  was  hung  on  a  horizontal  axis 
C,  and  the  reaction  measured  by  a  bent  lever 
balance  dDB,  upon  which  the  vessel  acted  by 
means  of  a  rod  GA,  bearing  on  the  point  directly 
opposite  to  the  orifice  F.  In  the  discharge 
through  an  orifice  in  the  thin  plate,  it  was  found 
that 

P=l,U  —  Fy. 

2£ 

If  we  take  the  section  of  the  stream:  _F,= 
0,64  .  F,  and  the  effective  velocity  of  discharge 
t>,  =  0,960  (Vol.  I.  §  315),  we  have,  according  to 
the  theoretical  formula  : 

P==2  .  !!>!,  F,  y  =  2  .0,96*  .  0,64  .  —  Fy 


Fig.  249. 


nearly  the  same  as  the  experimental  result.     When  the  orifice  was  provided  with  a 
mouth-piece  formed  like  the  vena  contracta,  it  was  found  that  : 
VOL.  II.  —  21 


242 


REACTION  WHEELS. 


P  =  1,73  .  _  F  y,  the  coefficient  of  discharge  being  0,94.     As  in  this  case  F ,  =  F,  and 

*g      ' 
r,  =  0,94  v,  the  theoretical  result  is : 

P  =  2.  0,94' £  *>=  1,77.  *.F» 
or  a  very  close  agreement  with  the  experimental  result. 

§  138.  Reaction  Wheels.— It  a  vessel,  as  HRF,  Fig.  250,  be 
placed  on  a  wheeled  carriage,  the  reaction  moves  the  carriage  in  the 
opposite  direction  from  that  in  which  the  discharge  takes  place,  and 
if  a  vessel  AF,  Fig.  251,  be  connected  with  a  vertical  axis  0,  it  will 


.  250. 


Fig.  251. 


cause  it  to  revolve  in  the  direction  opposite  to  that  in  which  dis- 
charge takes  place.  If  a  constant  supply  of  water  be  maintained, 
a  continuous  rotary  motion  results.  This  contrivance  is  the  reac- 
tion wheel  (Fr.  roue  a  reaction;  Ger.  Reactionsrad),  commonly 
called  Barker's  mill  in  Britain,  and  Segner's  water  wheel  in  Ger- 
many. The  simplest  form  of  this  wheel  is  shown  in  Fig.  252.  It 
consists  of  a  pipe  BC,  firmly  connected  with  a  vertical  axle  AX,  of 
two  pipes  CF  and  CGr  at  right  angles  to  the  first,  having  orifices  in 
the  sides  at  F  and  6r.  The  water  discharged  from  these  orifices  is 
continually  supplied  by  a  trough  leading  into  the  top  of  the  upright 


Fig.  252. 


Fig.  253. 


pipe.  In  applying  this  arrangement,  the  upper  millstone  is  gene- 
rally hung  immediately  on  the  axle  AX]  but  for  other  applications 
the  motion  might  be  transmitted  by  any  suitable  gear. 


REACTION  WHEELS. 


243 


Reaction  wheels  are  also  made  with  more  discharge  pipes  or  con- 
duits than  two,  as  shown  in  Fig.  253.  The  vessel  HR  is  made 
either  cylindrical  or  conical.  In  order  to  bring  in  the  water  at  the 
top  without  shock,  the  great  Euler  adapted  a  cylindrical  end  to  the 
pentrough,  immediately  above  the  wheel,  putting  a  series  of  inclined 
guide-buckets  into  it,  analogous  to  the  arrangement  introduced  by 
Burdin  for  his  turbines  (Vol.  II.  §  133). 

There  is  a  simple  reaction  wheel  erected  by  M.  Althans,  of  Val- 
lender,  in  the  neighborhood  of  Ehrenbreitstein,  for  driving  two  pair 
of  grindstones,  which  we  have  seen  and  admired.  The  arrange- 
ment of  this  machine  is  shown  in  the  accompanying  sketch,  Fig. 
254.  The  water  is  laid 

on  by  a  pipe  B  descend-  F »g-  2  54. 

ing  beneath  the  wheel, 
and  turning  vertically 
upwards.  The  upright 
axle  AC,  with  its  two 
arms  CF  and  (76r,  is 
hollow,  and  fits  on  to 
the  end  B  of  the  sup- 
ply pipe.  There  is  a 
stuffing  box  at  B  allow- 
_.  ing  of  the  free  rotary 
*>  motion  of  the  wheel, 
and  at  the  same  time 
preventing  loss  of  water 
at  the  joint.  The  rec- 
tangular orifices  F  and 
G-  are  opened  or  shut 
by  means  of  vanes  or 
slide  valves  moved  by 
rods  attached  to  a  collar 
E  on  the  axle,  movable  by  means  of  the  lever  EM.  The  water 
supplied  by  the  9  inch  pipe  B  flows  through  the  arms  of  the  wheel, 
and  through  the  apertures  F  and  6r.  This  arrangement  has  the 
advantage  of  supporting  the  whole,  or  great  part  of  the  weight  of 
the  machine  upon  the  water,  so  that  there  is  little  or  no  friction  on 
the  base.  If  Cr  be  the  weight  of  the  machine,  h  the  head  of  water, 
2  r  the  diameter  of  the  pipe  at  B,  then  *  r2  h  v  =  6r,  and,  therefore, 
in  order  to  support  the  machine,  the  radius  of  the  pipe  should  be 

r  =     I— — .     The  quantity  of  water  expended  by  this  machine  is 

\  *  Ay 

18  cubic  feet  per  minute,  the  fall  is  94  feet,  and,  therefore,  the 
mechanical  effect  at  disposition  is  1755  feet  Ibs.  per  second.  The 
length  of  the  arms  is  12  J  feet,  and  the  number  of  revolutions  30  per 
minute,  or  the  velocity  at  the  periphery  39,3  feet  per  second. 

Remark  1.  The  first  account  of  a  reaction  wheel,  as  an  invention  of  Barker,  is  given 
in  Desagulier's  "  Course  of  Experimental  Philosophy,  vol.  ii.  London,  1 745."  Euler  treats 


244  THEORY  OF  THE  REACTION  WHEEL. 

in  detail  of  the  theory  and  best  construction  of  these  wheels  in  the  "  Memoirs  of  the 
Berlin  Academy,  1750—1754." 

Remark  2.  The  efficiency  of  reaction  wheels  is  reputed  as  extremely  small.  Nord- 
wall  makes  it  only  half  of  that  of  an  overshot  wheel,  and  Schitko's  experiments  on  such 
a  wheel  gave  the  efficiency  only  0,15. 

§  139.  Theory  of  the  Reaction  Wheel—  The  effects  of  reaction 
•wheels  may  be  determined  theoretically  as  follows.  If  h  be  the 
fall,  or  the  depth  of  the  centre  of  the  orifices  below  the  surface  in 
the  feed-pipes,  we  have  the  height  measuring  the  pressure  of  water 

on  the  orifices  Ax  =  h  -f-  —  ,  and,  therefore,  the  theoretical  velocity 

of  discharge  c  =  t  \/2gh  +  v2.  This  is  not,  however,  the  absolute 
velocity  of  the  water  at  efflux  from  the  wheel,  for  it  partakes  of  the 
velocity  of  rotation  v  in  common  with  the  wheel,  in  the  opposite 
direction.  Hence  the  absolute  velocity  of  the  water  leaving  the 
wheel:  w  =  c  —  v  =  <j>  \/2gh  +  v3  —  v,  and  the  loss  of  mechanical 
effect  corresponding: 

J,A  ^  w*  Q  r  -  (*  SW+^Z~  v?  n  y, 
The  co-efficient  of  velocity  $  being  assumed  =  1,  then 


.  . 

and  deducting  this  from  the  effect  at  disposition,  the  useful  effect 

remaining  is  : 


_  (h      w* 
\        2 


9 
This  increases  as  v  increases,  for  if  we  put  : 

—  £*- 

v         2  v 


n  r. 


2  v3  9 

and  for  v  =  oo  ,  L  =  Q  h  y,  the  whole  effect  available. 

This  circumstance  of  the  maximum  effect  depending  on  the  wheels 
acquiring  an  infinite  velocity,  is  very  unfavorable  ;  because,  as  the 
velocity  increases,  the  prejudicial  resistances  increase,  and  even 
when  the  wheel  runs  without  any  useful  resistance,  the  velocity  it 
acquires  is  far  from  being  infinite,  proving  the  absorption  of  effect 
at  these  great  velocities. 

The  question  of  course  is,  as  to  whether  the  effect  for  mean  velo- 
cities of  rotation  be  very  much  less  than  the  maximum  effect,  or 
Q  h  y.  If  we  load  the  machine  to  such  an  extent  that  the  height 
due  to  velocity,  corresponding  to  the  velocity  of  rotation,  is  equal 

to  the  fall,  or  —  =  h,  or  v  =  <>/2gh,  then,  according  to  the  above 
formula: 


WHITELAW'S  TURBINE. 


245 


but  if  ~ 


,  then: 


and,  lastly,  if  —  =  4A,  then: 


_1}  QAy_0>899  QJr- 


Qy= 


It  thus  appears  that,  in  the  first  case,  we  lose  17,  in  the  second  10, 
and  in  the  third  only  6  per  cent,  of  the  available  effect,  and,  there- 
fore, for  moderate  falls,  and  when  a  velocity  of  rotation  exceeding 
the  velocity  due  to  the  height  of  fall  may  be  adopted,  there  is  a 
great  effect  to  be  expected  from  these  wheels.  Considering  the 
great  simplicity  of  these  wheels,  the  balance  must  often  be  much  in 
their  favor  when  compared  with  other  wheels. 

Remark.  The  force  of  rotation  or  of  reaction  is  : 

v 


Fi«-  255- 


y==f.Qy  =  2  .  —  F  y,  as  we  showed,  Vol.  II.  §  146,  al- 
g  g  2£ 

though  by  a  different  method. 

§  140.  Whitelaw's  Turbine.  —  Within  the  last  few  years,  the  pipes 
or  conduits  of  reaction  wheels  have  been  made  curved,  and  such 
wheels  are  known  as    Whitelaw  '«,  or  Scottish  turbines.     Manouri 
d'Ectot  constructed  wheels  on  nearly  the  same  plan  as  long  ago  as 
1813,  (see  "Journal  des  Mines,  t.  xxxiii.")     The  Scottish  turbines, 
constructed  by  Messrs.  Whitelaw  and  Stirrat,  of  Paisley,  are  de- 
scribed in  the  "  Description  of  Whitelaw  and  Stirrat's  Patent  Water 
Mill,  2d  edition,  London  and  Birmingham,  1843."     One  peculiarity 
of  Whitelaw's  wheel  consists  in  the  in- 
troduction   of  a   movable  piece  at  the 
outer  orifice  of  the  conduits  by  which  its 
section  is  regulated.     Fig.  255  is  a  hori- 
zontal section  of  one  of  Whitelaw's  tur- 
bines.    There   are,  in   this   case,  three 
arms.     The  water  enters  at  E,  and  is 
discharged  at  A.     OA  is  a  valve  mova- 
ble round  0,  by  which  the  orifice  of  dis- 
charge  is   regulated.     The   adjustment 
of  these  regulators  has  been  made  self- 
acting  by  a  peculiar  arrangement,  but 
is  better  regulated  by  the  hand,  by  an 
apparatus  analogous  to  that  shown  in  Fig.  254. 

The  general  arrangement  of  Whitelaw's  turbines  is  clearly  shown 

by  Fig.  256.     A  is  the  lead  for  the  water.     B  the  sluice.     C  a 

reservoir  immediately  above  the  pressure  pipe.     E  is  a  valve  for 

regulating  the  expenditure  of  water.     At  F  the  water  enters  the 

21* 


246 


COMBE  S  REACTION  WHEEL. 


cylinder  6r,  and  goes  from  thence  into  the  wheel  HK  placed  above 
it,  and  fixed  on  the  vertical  axis  LM.     The  reaction  of  the  water 


Fig.  256. 


streaming  from  the  three  orifices,  drives  the  wheel  round  in  the 
opposite  direction,  and  this  motion  is  transmitted  by  the  bevelled 
gear  LN.  The  wheel,  the  axle,  and  the  pressure  pipe  are  of  cast 
iron.  The  footstep  for  the  pivot  at  M  is  of  brass.  Oil  is  intro- 
duced by  a  pipe  0  from  the  wheel  room. 

We  shall  hereafter  recur  to  the  theory  and  geometrical  construc- 
tion of  this  wheel. 

§  141.  Combe  s  Reaction  Wheels. — As  being  analogous  to  White- 
law's,  we  may  next  consider  Combe's  reaction  wheel.  The  water 
flows  from  below  upwards  into  these  wheels,  and  the  wheel  differs 
essentially  from  Whitelaw's  in  having  so  great  a  number  of  conduits 
or  orifices  of  discharge,  that  it  may  be  said  to  discharge  at  every 
point  of  the  circumference,  as  the  plan  of  the  wheel  in  Fig.  257 
shows.  AA  is  a  plate  connected  with  the  axis,  and  forming  the 
upper  shrouding  or  cover  of  the  wheel.  SB  is  the  under  shrouding, 
and  upon  it,  between  this  and  the  upper  plate,  the  buckets  EE  are 
fastened.  DD  is  a  cylinder  surrounding  the  lower  part  of  the  axis, 


CADIAT'S  TURBINE. 


247 


through  which  the  water  is  laid  on  to  the  wheel,  entering  the  wheel 

by  all  the  apertures  between  the  buckets  on  the  inside,  and  stream- 

ing through  the  conduits  formed  by  the  buckets  to  be  discharged 

at  every  point  of  the  outer  circumference.     Another  essential  differ- 

ence between  this  and  Whitelaw's  wheel  arises  from  the  absence  of 

the  water-tight  joint  between  the  wheel 

B  and  the  end  of  the  pipe  leading  the 

water  to  the  wheel,  and  which  is  quite 

necessary  to  Whitelaw's  wheels.     The 

reason  of  this  difference  .  is,  that  the 

pressure  of  water  in   a  reservoir  or 

vessel,  from  which  the  water  is  run- 

ning, is  different  at  different  points. 

The  pressure  is   greatest  where   the 

water  is  nearly  still,  and  least  where  the 

velocity   is  greatest  (Vol.  I.  §  307). 

The  velocity  of  the  water   depends, 

however,  upon  the  section  of  the  reser- 

voir or  vessel,  and  is  inversely  propor- 

tional to   the   section,   and   thus,  by 

varying  the  section,  the  pressure  may 

be  made  to  vary  at  will,  or  it  may  be 

made  only  equal  to  that  of  the  atmo- 

sphere.    If  we  bore  a  hole  in  the  side 

of  a  vessel  at  the  point  where  the  pressure  of  the  water  flowing  past 

is  only  equal  to  the  atmospheric  pressure,  there  will  be  no  discharge, 

nor    any   indraft.      Thus,    that   no 

water  may   escape,    and  no  air   be 

drawn  in  through  the  space  neces- 

sarily left  between  B  and  D,  it  is 

only  necessary  to   give  the  section 

certain  dimensions  at  the  point  of 

passage. 

Remark.  Combe's  wheels  are  sometimes  pro- 
vided with  guide-buckets  for  laying  the  water 
on  to  the  wheel  in  a  definite  direction. 

Redtenbacher  forms  the  water-tight  joint  be- 
tween the  main  pipe  JIB  and  the  wheel  DEF, 
by  means  of  a  movable  brass  ring  CD,  which 
is  pressed  so  tightly  up  against  the  lower  ring- 
surface  D  of  the  wheel,  that  the  water  does 
not  escape.  The  ring  CD  must  slide  in  a  well- 
constructed,  water-tight  collar. 

§  142.  Cadiat's  Turbine.—  The  next  wheel  we  shall  describe  is 
Cadiat's  turbine.  These  have  no  guide-curves,  like  Whitelaw's  and 
Combe's,  but  as  in  Fourneyron's  turbine,  the  water  is  brought  in 
from  above.  The  peculiarity  of  this  wheel  is  the  introduction  of  a 
cylindrical  sluice,  which  shuts  the  wheel  on  the  outside.  Fig.  259 
is  a  vertical  section  of  this  wheel.  AA  is  the  wheel,  BE  being  a 
saucer-formed  plate  connecting  the  wheel  with  the  axle  CD.  The 


Fig.  258. 


248 


FOURNEYRON'S  TURBINE. 


pivot  C  of  this  axle  rests  in  a  footstep,  which  we  shall  hereafter 
have  occasion  to  allude  to  more  particularly.  EE  is  the  reservoir 
with  circular  section  communicating  with  the  lead  W,  and  coming 

Fig.  259. 


in  immediate  contact  with  the  upper  plate  or  shrouding  of  the  wheel. 
That  the  water  coming  into  the  wheel  may  not  be  unnecessarily 
disturbed  or  contracted,  the  reservoir  gradually  widens  both  upwards 
and  downwards,  as  the  figure  shows.  The  discharge  of  the  water  is 
regulated  by  the  cylindrical  sluice  FF  on  the  outside.  This  sluice 
is  raised  or  lowered  by  3  or  4  rods,  connected  with  mechanism  for 
the  purpose.  That  there  may  be  no  escape  of  water  between  the 
sluice  and  the  side  of  the  reservoir,  the  joint  is  made  with  leather. 

The  upright  axle  CD  is  enclosed  in  a  pipe  HH,  to  the  bottom  of 
which  is  attached  a  plate  KK,  reaching  to  the  inner  circumference 
of  the  lower  shrouding  of  the  wheel,  so  that  the  water  is  shut  off 
from  the  disc  or  plate  BB  of  the  wheel.  This  arrangement  is  adopted 
from  that  first  introduced  by  Fourneyron  in  his  turbine. 

Remark.  A  complete  and  accurate  description  of  one  of  Cadiat's  turbines,  as  originally 
constructed,  is  given  by  Armengaud,  sen.,  in  the  second  volume  of  his  "  Publication  In- 
dustrielle." 

§  143.  Fourneyron's  Turbine. — Fourneyron's  turbines,  as  they 
have  been  recently  made,  may  be  considered  as  among  the  most 
perfect  horizontal  wheels.  They  work  either  in  or  out  of  back-water, 
are  applicable  to  high  and  to  low  falls — are  either  high  pressure  and 
or  low  pressure  turbines.  In  the  low  pressure  turbine,  the  water 
flows  into  the  reservoir,  open  above,  as  shown  in  Fig.  260.  In  high 
pressure  turbines,  the  reservoir  is  shut  in  at  top,  and  the  water  is 
laid  on  by  a  pipe  at  one  side,  as  represented  in  Fig.  261.  The 
wheel  consists  essentially  of  two  shroudings,  between  which  are  the 
buckets  of  the  connecting  plate  or  arms,  and  the  upright  axle,  as  in 


FOURNEYRON'S  TURBINE.  249 

Cadiat's  turbine.  The  water  from  the  lead  W  flows  into  the  reser- 
voir EE.  In  order  that  the  water  may  not  rest  directly  on  the 
wheel  disc  BB,  which  would  greatly  increase  the  pivot  friction,  a 
pipe  encloses  the  upright  axle,  to  which  there  is  attached  a  disc  FF, 

Fig.  260. 


250 


FOURNEYRON  S  TURBINE. 


upon  which  the  mass  of  water  presses  as  it  flows  to  the  wheel.  On 
this  disc  the  so-called  guide-curves,  ab,  afi^  &c.  (Fig.  260  or  262),  are 
fastened.  These  give  the  water  a  certain  direction  of  motion  on  to 
the  wheel,  which  surrounds  them,  and  through  it  along  the  buckets, 
to  he  discharged  at  the  outer  circumference.  The  reaction  is  such 


that  the  wheel  revolves  in  the  opposite  direction,  the  guide-curves 
and  their  support  remaining  at  rest.  To  regulate  the  discharge  of 
water,  there  is  a  cylindrical  sluice  KLLK,  Fig.  262,  in  the  inside, 
which  is  raised  and  depressed  hy  three  rods  M,  M .  .  .  ,  connected 
with  mechanism  suited  to  the  object.  The  sluice  KL  is  a  hollow 
cast  iron  cylinder,  the  outer  surface  of  which  nearly  touches  the 
inside  of  the  upper  shrouding  of  the  wheel,  and  they  must,  there- 
fore, be  both  accurately  turned.  The  sluice  is  made  to  slide  water- 
tight in  the  reservoir  by  a  leather  or  other  fitting  above  LL.  The 
sluice  is  generally  lined  with  wood,  rounded  at  top  and  bottom,  so 
as  to  prevent  contraction  as  much  as  possible,  that  is,  to  prevent 
all  losses  of  vis  viva.  In  high  pressure  turbines,  the  rods  for  work- 
ing the  sluice  pass  through  the  cover  of  the  reservoir  through  stuff- 
ing boxes.  Sometimes  the  regulation  is  effected  by  raising  or  de- 


FOURNEYRON'S  TURBINE. 


251 


pressing  the  bottom  plate  Fig.  263.  For  this  purpose,  the  top  of 
the  pipe  Crffis  screwed,  and  the  female  screw  Mis  attached  to  a 
conical  wheel,  moved  by  gear,  as  0.  The  female  screw  is  fixed  so 

Fig.  262. 


252 


THE  PIVOT  AND  FOOTSTEP. 


that  its  motion  raises  or  lowers  the  pipe  GrH.     There  is  also  attached 
to  Gr H  a  plate  or  piston  ffL,  having  a  water-tight  packing. 


§  144.  The  Pivot  and  Footstep. — The  pivot  and  footstep  are  very 
important  parts  of  the  turhine.  The  weight  of  the  turbine,  often 
considerable,  and  the  velocity  of  rotation,  give  rise  to  a  great  moment 
of  friction  on  the  pivot,  which  would  wear  very  rapidly,  unless  it 
were  well  greased.  It  has  been  frequently  observed  that  the  pivot 
and  brass  of  turbine  axles  wear  much  more  rapidly  than  the  pivots 
of  other  upright  axles.  This  is  attributable  partly  to  the  heating  of 
the  pivot  from  great  velocity  of  rotation,  but  chiefly  to  the  difficulty 
of  lubricating  the  bearing-points  which  are  under  water.  In  order 
to  meet  this  evil,  turbine  makers  have  endeavored  to  diminish  the 
weight  as  much  as  possible,  to  increase  the  rubbing  surface,  to  pre- 
vent the  contact  of  the  water  with  the  rubbing  surfaces,  and  also,  to 
keep  up  a  continuous  supply  of  olive  oil  or  nut  oil,  between  the  sur- 
faces in  contact. 

At  the  upper  end  of  the  axle,  there  must  of  course  be  a  collar  or 
other  support,  in  which  it  can  revolve. 

A  very  simple  footstep,  applicable  chiefly  when  the  weight  is 
inconsiderable,  is  shown  in  Fig.  264.  The  pivot  0  rests  in  a  brass 


THE  PIVOT  AXD  FOOTSTEP. 


253 


D,  which  is  supported  on  a  block,  movable  up  or  down,  as  may  be 
required,  by  means  of  the  folding  key  PS.     The  oil  is  supplied  by 

Fig.  264. 


Fig.  265. 


a  pipe  R  passing  by  the  side  of  the  key,  and  through  the  bottom  of 
•  the  block  and  brass. 

Fig.  265  is  the  arrangement  of  footstep  adopted  by  Cadiat.  A 
the  foot  of  the  upright  shaft.  B  a  hardened 
steel  pivot  attached  to  A  by  screw  or  weld- 
ing. 0  is  a  hardened  steel  step  for  the  pivot. 
DEED  is  a  cast  iron  block  or  case  for  the 
step.  EE  being  a  brass  casing.  F  is  a  pipe 
taking  oil  to  the  space  between  B  and  E.  Gr 
is  a  lever  for  raising  or  depressing  the  tur- 
bine. 

Fourneyron  has  very  much  complicated  the 
construction  of  the  pivot  and  footstep,  to  attain 
permanence.  The  general  arrangement  is 
shown  in  Fig.  262,  and  its  details  are  shown 
in  Fig.  266.  Fig.  262  shows  that  the  footstep  Z  is  in  a  block  which 
rests  on  a  lever  OR,  turning  round  0,  when  elevated  or  depressed 
by  the  rod  RS.  U  is  a  pipe  for  bringing  oil  to  the  footstep,  the 
head  of  which  should  be  as  high  as  possible.  That  the  circulation 
of  the  oil  may  be  active,  it  should,  at  all  events,  be  considerably 
above  the  surface  of  the  water  in  the  lead  in  low  pressure  turbines, 
and  there  should  be  a  means  of  forcing  in  a  supply  for  high  pressure 
turbines.  The  parts  A  and  B  immediately  in  contact  with  each 
other  are  of  hard  steel.  The  upper  part  A  is  fixed  in  the  shaft  Gr, 
and  the  lower  part  B  fits  into  a  hollow  piece  DD,  and  is  movable 
upwards  and  downwards  by  a  lever  supporting  the  whole,  and 
passing  through  0  (OR,  Fig.  262).  The  surface  of  A  is  hollowed, 
VOL.  n. — 22 


254 


THEORY  OF  REACTION  TURBINES. 


and  the  head  of  B  rounded,  and  both  are 
surrounded  by  a  collar  EE,  which  keeps  the 
oil  between  the  rubbing  surfaces.  The  oil, 
brought  down  in  a  pipe,  enters  at  a  into  the 
hollow  space  5,  and  from  thence,  through  c, 
passes  into  the  space  d.  Out  of  this  it  flows 
through  three  channels  ef,  on  the  periphery 
of  the  steel  bearing,  rising  perpendicularly 
from  the  bottom  and  running  inclined  to  the 
top,  where  three  radial  furrows  serve  to  dis- 
tribute it  sufficiently.  Lastly,  there  goes  from 
the  centre  of  A,  a  hole  gh  into  the  axis, 
through  which  the  oil  escapes  outwards,  so 
that  a  circulation  is  maintained. 

§  145.  Strength  or  Dimensions. — In  de- 
signing a  turbine  for  a  certain  fall  of  water, 
there  is,  besides  the  chief  dimensions  of  the 
wheel  itself,  the  strength  of  certain  parts  to  be 
calculated.  The  strength  of  pipes,  &c.,  is  to  be  calculated  by  the  for- 
mulas given  in  Vol.  I.  §  283;  and  at  Vol.  II.  §  84,  we  have  treated 
of  the  dimensions  necessary  for  shafts.  If  L  be  the  useful  effect 
of  a  turbine  in  horse's  power,  and  u  the  number  of  revolutions 
per  minute,  we  have  for  the  requisite  diameter  of  the  upright  shaft 

[ —  inches. 

M 

The  strength  of  the  bottom  plate,  &c.,  is  easily  determined  by 
reference  to  the  theory  of  the  strength  of  materials;  but  the  nature 
of  the  casting  fixes  the  dimensions,  so  that  there  can  very  rarely 
occur  any  necessity  for  calculation. 

Remark  1.  In  the  erection  of  turbines,  not  only  the  weight  of  the  parts  of  the  machine, 
but  the  water  pressure,  has  to  be  considered.  The  latter  has  especially  to  be  considered 
in  high  pressure  turbines.  It  must  not,  for  example,  be  lost  sight  of,  that  the  water 
presses  against  the  reservoir  with  a  force  equal  to  the  weight  of  a  column  of  water 
having  the  section  of  the  pressure  pipes  as  base,  and  the  head  of  water  as  height ;  and 
that  the  knee  piece  on  the  pressure  pipes  tends  to  move  with  the  same  force,  but  in  the 
opposite  direction. 

Remark  2.  For  determining  the  dimensions  of  the  pivot,  the  rule  which  limits  the 
pressure  on  every  square  inch  of  the  brass  to  1500  Ibs.,  and  that  for  a  steel  pivot  on  a 
steel  bearing  to  7000  Ibs.,  might  be  used ;  but  we  know  from  what  has  been  said  above 
in  reference  to  the  wear  of  these  points,  that  it  is  preferable  to  make  the  pivot  only  very 
little  less  than  the  diameter  of  the  shaft.  The  above  numbers  refer  to  shafts  having  a 
moderate  velocity  of  rotation,  and  as  the  wear  increases  as  the  weight,  and  as  the  velo- 
city of  rotation  besides,  it  is  evident  that  turbines,  revolving  with  great  speed,  should 
have  proportionally  large  pivots  and  bearings. 

§  146.  Theory  of  Reaction  Turbines. — For  the  investigation  of 
the  mechanical  proportions  and  effects  of  Fourneyron's  turbines,  we 
shall  use  the  following  notation : 

rl  —  CA,  Fig.  267,  the  radius  of  the  inside  of  the  wheel,  or  ap- 
proximately, that  of  the  periphery  of  the  bottom  plate. 

r  =  CB,  the  radius  of  the  outside  of  the  wheel. 

i\  =  the  velocity  of  the  interior  periphery. 


THEORY  OF  REACTION  TURBINES. 


255 


v  =  the  velocity  of  the  exterior  Fig-  267. 

periphery. 

c  =  the  velocity  with  which  the 
water  flows  from  the  reservoir  or 
guide-curves. 

Cj  =  the  velocity  with  which  the 
water  enters  the  wheel. 

c2  =  the  velocity  with  which  the 
water  leaves  the  wheel. 

a  —  the  angle  cA  T  which  the 
direction  of  the  water  leaving  the 
reservoir  makes  with  the  inner  cir- 
cumference of  the  wheel. 

j3  =  the  angle  c^A  T  made  by  the 
water  entering  the  wheel  buckets 
with  the  inner  periphery  of  the 
wheel. 

8  =  the  angle  c2A  T  made  by  the  water  stream  leaving  the  wheel 
with  the  outer  periphery. 

F=  the  area  of  the  orifices  of  discharge  from  the  guide-curves. 

Fl  —  the  sum  of  the  areas  of  the  orifices  by  which  the  water  enters 
the  wheel. 

F2  =  the  sum  of  the  areas  of  the  orifices  at  the  outside  of  the 
^buckets. 

h  =  the  entire  fall  of  water. 

Aj  =  the  height  from  surface  of  lead  to  centre  of  wheel,  or  cen- 
tre of  orifice  of  discharge  from  reservoir. 

h2  =  hl  —  h  the  depth  of  the  entrance  orifices  to  the  wheel,  below 
the  orifices  of  discharge,  or,  if  the  wheel  works  under  water,  below 
the  surface  of  the  tail-race;  and,  lastly: 

x  =  the  height,  measuring  the  pressure  of  the  water  at  the  point 
where  it  passes  from  the  reservoir  or  guide-curves  into  the  wheel. 

In  the  first  place,  for  the  velocity  e  due  to  the  difference  of  pres- 
sures h1  —  x,  we  have  —  =  hl  —  x,  or,  more  accurately,  if  the  water, 
in  flowing  from  the  guide-curves,  loses  an  amount  of  hydraulic  pres- 
sure =  I  .  — ,  then  (1  +  1}  —  =  Ax  —  x,  therefore, 

t/ t/ 

if),  and,  inversely,  x  =  A,  -  (1  +  $  £-. 

In  order  that  the  water  may  enter  the  wheel  without  shock,  it  is 
necessary  that  the  velocity  of  discharge  should  resolve  itself  into 
two  components,  the  one  of  which  must  coincide  in  magnitude  and 
direction  with  the  velocity  of  the  inner  circumference  of  the  wheel, 
and  the  other  must  coincide  in  direction  with  the  stream  entering 
the  wheel  conduits  or  channels.  This  being  taken  for  granted:  if 
the  velocity  with  which  the  water  begins  to  flow  through  the  chan- 
nels Ac^  =  <?1?  we  have  it  from  the  formula  : 

C?  =  C2  +  V,2  —  2  C  Vl  COS.  a. 


256  BEST  VELOCITY. 

The  velocity  of  discharge  c2  may  be  deduced  from  the  pressure 
height  x  and  Ti2  at  entrance  and  exit,  from  the  height  JL ,  being  that 

corresponding  to  the  velocity  of  entrance,  and  from  the  increase  of 
pressure  height  corresponding  to  centrifugal  force  of  the  water  in 

the  wheel!— A2  (Vol.  II.  §  143): 

^    c*       *_,. 
^-=rc  —  A2  +  -L  -1 — L ,  or,  substituting  the  above  values  of 

x  and  cl : 


.l_,__   _   _ 

or  as  Aj  —  h2  =  A,  the  whole  fall  : 


2  <?  v1  cos.  a  — 


.  <?2. 
rvili 


If  we  further  assume  that,  by  friction  and  curvilinear  motion  in 
the  wheel  channels,  there  is  a  loss  of  hydraulic  head  =  ^-£?_,  then, 
more  accurately  :  (1  +  *)  £22  =  2$rA  -f  #2  —  2  c  v1  cos.  a  —  f  .  c3. 

The  quantity  of  water  Q  =  Fc  =  F1cl  =  F2c2  .'.  c  =  ±p  ,    and 

F 

Vi  =  r-±  v,  and,   therefore,  we  have  for  the  velocity  of  the  water 
leaving  the  wheel: 

2  +  2        "  v  C08'  a~v*=  2- 


§  147.  Best  Velocity.  —  In  order  to  get  the  maximum  effect  from 
the  water,  the  absolute  velocity  of  the  water  leaving  the  wheel  must 
be  the  least  possible.  But  this  velocity  is  the  diagonal  Bw  of  a 
parallelogram  constructed  from  the  velocity  of  discharge  c2  and  the 
velocity  of  rotation  v: 

w  =  -v/c22  +  v3  —  2  c2  v  cos.  S  =     i(c2  —  v)2  +  4  c2  v  (sin.  -\  ; 

and  S  is  to  be  made  as  small  as  possible,  and  c2  =  v.  But  in  order 
that  there  may  be  free  passage  for  the  necessary  quantity  of  water, 
it  is  not  possible  to  make  8=0,  but  this  has  to  be  made  10°  to  20°  ; 
whenever,  therefore,  we  make  cz  =  v,  there  remains  the  absolute 
velocity:  _ 

"2' 

and  the  loss  of  effect  corresponding  is: 


2  2,  y. 

We  now  perceive  that  the  maximum  effect  is  not  got  when  v  =  cv 
but  when  v  is  something  less  than  c2  ;  but  it  is  also  manifest  that 
for  v  =  cv  and  for  a  small  value  of  8,  the  deficiency  below  the 


PRESSURE  OF  THE  WATER.  257 

maximum  effect  can  only  be  very  small.  As,  besides,  in  assuming 
v  =  c2we  get  very  simple  relations,  we  shall  do  so  in  the  sequel, 
and  introduce  this  into  the  last  equation  of  the  preceding  paragraph. 
We  then  have: 


]  v*  +  2  T  '  ~  v*  C08'  a  —  v2==  23h>  or' 


7"  C08'  a  +  ^  "       +  *    v2=  2gh'  and'  therefore'  the  vel°- 

city  of  the  wheel  corresponding  to  the  maximum  effect  required,  is  : 


-n 

Instead  of  the  section  ratio     *   we   may  introduce  the   angle  3. 

M 

The  unimpeded  entrance  of  the  water  into  the  wheel  requires  that 
c  should  not  be  altered  in  entering  into  it,  or  that  the  radial  com- 
ponent of  c,  AN  =  c  sin.  o,  should  be  equal  to  the  radial  component 
of  cv  i.  e.  cl  sin.  3,  and  also  to  the  tangential  component  c  cos.  a.  of 
c  the  tangential  velocity  A  T  =  cl  cos.  3  +  v1  of  the  water  already 
within  the  wheel.  According  to  this  : 

c.       sin.  a  ,   c          sin.  3 

-,  c  cos.  a  —  cl  cos.  3  =  vv  and  —  = 


>  c       sin.  3  vl       sin.  (3 — o) 

as  Fc  =  FyC2  =  F  v  =  —  F2v1 ;  we  have 

ri 
F       r,      c  .     r.         sin.  3 


F       r  '  vl       r  '  sin.  (3  —  a)' 
and  the  velocity  of  the  outer  periphery  of  the  wheel : 


2  (rJL\2  8in'  fl  C08'  a   |   ?  (     r*  8in' fl 
V  r  )  sin.  (3  —  a)          \r  sin.  (3  —  o 

and  hence  the  velocity  of  the  inner  periphery : 


in.  3  cos,  a       .  /       sin.  8        \2          /r\* 
in.  (3  —  a)  "*"     \r  sin.  (3  —  a)/        *  \r,  / 


__tang.  vcotang.fi). 

.  3  C08.  a 


sin.  3  cos.  a 

sin. 

Neglecting  the  prejudicial  resistances,  we  should  have : 
_     \gh  sin.  (s  —  o) 


§  148.  Pressure  of  the  Water.  —  With  the  aid  of  this  formula  for 
w,  we  can  determine  the  pressure  exerted  on  that  part  of  the  reservoir 
where  the  water  passes  from  it  on  to  the  wheel  ;  we  have  : 


22* 


258  PRESSURE  OF  THE  WATER. 


(1  +  f)  h  sin.  ft2 


2  sin.  ft  cos.  a  sin.  (ft  —  o)  +  f  *m.  /32  +  *  (~\   [sin.  (ft  —  a)]2 

Wj/ 


=  A,— 


-i     ,  o  „      •       c,  r  /r\*/Sin.  (3 a) 

1  +  cos.  2  a  —  cotg.  ft  sin.  2a  +  £-fx(_)  ( S '- 

\r1/   \      sin.  ft 
Neglecting  prejudicial  resistances,  we  have : 


a;  =    ,  - 


cos.  2  a  —  cotg.  ft  sin.  2  a 
If  the  turbine  work  free  of  back  water,  we  have,  in  the  case  of 
the  turbines  of  Fourneyron,  Cadiat,  and  Whitelaw,  hl  =  h,  and, 
therefore, 

cos.  2  o  —  cotg.  ft  sin.  2  o         7  . 
1  +  cos.  2  a  —  cotg.  ft  sin.  2  o 

if,  however,  the  turbine  works  in  back  water,  then  h1  =  h  +  h2,  and, 
therefore, 

cos.  2  a  —  cotg.  ft  sin.  2  a        7,7 

x  =  ^ — 2 . —      .  h  +  h2. 

1  +  cos.  2  o  4-  £0^7-  ft  sin.  2  a 

If,  in  the  first  case,  the  pressure  is  to  be  =  0,  i.  e.,  equal  to  the 
atmospheric  pressure,  then  x  =  0 ;  but  if  in  the  second  case,  it 
must  be  equal  to  the  pressure  of  the  back  water  against  the  orifices 
of  the  wheel,  then  x  =  A2,  but  in  both  cases  we  should  have : 
cos.  2  a  —  cotg.  ft  sin.  2  a  =  0,  i.  e.,  tang,  ft  =  tang.  2  o,  or  ft  =  2  o. 

If,  therefore,  the  angle  of  the  water's  entrance  ft  be  twicet  he  angle 
of  exit  a,  the  pressure  at  the  point  where  the  water  parses  from  the 
reservoir  to  the  wheel,  is  equal  to  the  external  pressure  of  the  atmo- 
sphere, or  of  the  back  water. 

On  the  other  hand,  it  is ,  easy  to  perceive  that  this  internal  pres- 
sure is  greater  than  the  external  pressure,  if  ft  >  2  o,  and  it  is  less 
than  this  when  ft  <  2  a.  These  relations  are  somewhat  different 
when  the  prejudicial  resistances  are  taken  into  account,  as  it  is  very 
proper  to  do.  The  equation  between  the  external  and  internal  pres- 
sure then  stands  thus : 

1  +  C08.  2  0  —  cotg.  ft  sin.  2  «  4-  £  4-  «  f-V  i8™'1*  —  a)\2  =  1  4-  ?, 

\rt/  \     sin.  ft     J 

(r  \2 
—  J  (cos.  a  —  cotg.  ft  sin.  o)2. 

If,  in  the  last  member,  we  introduce: 

sin.  2  a ' 
that  is : 

cotg.  ft  sin.  2  a  =  cos.  2  o  +  x  (—\*  (  8m"  °  V* 
Vr,  /  \sin.  2  a  / 


it  follows  that: 


1 

rj   '  4  (cos.  a)2' 


tang.  ft 


CHOICE  OF  THE  ANGLES  a  AND  ft.  259 

_  sin.  2  a 

COS.  2a+  xl-2 


4  (COS.  a)3 

consequently,  j3  is  somewhat  smaller  than  2  o. 

If  we  neglect  again  £  and  x,  we  have,  by  introducing  the  value 
ft  =  2  a. 


^ 
|S 

\ 


, 

2  COS.  a 

and  c  =  \/2gh.     If  the  internal  pressure  be  greater  than  the  ex- 

ternal, then  v1  >  S_IiL,  and  c  <  \S2gh,  and  if  it  be  less  than  this, 
cos.  a 

then  v,  ^  V?gh,  and  c  =>  ^/<p. 

COS.  a 

§  149.  The  discussion  as  to  pressures  in  the  last  paragraph  is 
of  great  importance  in  the  question  of  the  construction  of  turbines; 
for  the  point  of  transit  from  the  reservoir  to  the  wheel  cannot  be 
made  water-tight,  and,  therefore,  water  may  escape,  or  water  and 
air  get  in,  by  the  annular  aperture.  That  neither  of  these  circum- 
stances may  occur,  turbines  must  be  so  constructed,  that  the  inter- 
nal pressure  of  the  water  passing  the  slit  may  equal  the  external 
pressure  of  the  atmosphere,  or  of  the  back  water,  if  the  wheel  be 
submerged  ;  we  must,  in  short,  have  /3  =  2  o,  or,  better  still,  satisfy 
the  equation: 

sin.  2  a 
tang.  ft  = 


cos.  2  o  + 


& 


(2  COS.  a)2 

Turbines  are  constructed  so  that,  in  the  normal  state  of  the  sluice 
being  fully  opened,  the  above  equation  is  satisfied,  or,  so  that  a 
slight  excess  of  pressure  x  exists,  at  the  risk  of  losing  some  water 
through  the  space  left  between  the  bottom  plate  and  the  wheel,  thus 
providing  for  variations  of  velocity  of  discharge,  from  variations  in 
the  area  of  the  section  F  of  the  orifices  by  the  different  positions  of 
the  turbine-sluice,  regulating  the  quantity  discharged. 

§  150.  Choice  of  the  Angles  o  and  j3. — If  we  do  not  take  the  in- 
ternal pressure  into  consideration,  the  angles  a  and  ft  may  have  very 
arbitrary  values.  The  formula 


(1  -  tang,  a  cotg.  ft)  =  ^gh  (l  _ 


gives  an  impossible  value  for  vv  when  — £l?  >  1,  that  is,  when 

tang.  ,3 

0  <  90°,  and  ft  <  a,  or  when  a  >  90°,  and  ft  >  o.  These  values  of 
a  and  ft  are,  therefore,  not  to  be  admitted.  If  o  =  ft,  then  v^  =  0, 
and  hence  we  see  that  the  best  velocity  of  rotation  becomes  so  much 
the  less,  the  nearer  the  angles  o  and  ft  approach  equality.  The 

formulas  c  =  _^1^_,  and.F2  =  !i  .  _J*L'J_-1T,  given  negative 
sin.  (ft  —  a)  r     sin.fi  —  ») 


260  CHOICE  OF  THE  ANGLES  o  AND  ft. 

values,  involving  impossible  conditions,  when  ft  >  0.     It  is,  there- 
fore, necessary,  in  the  construction  of  turbines,  that  ft  >  0,  and  that 

a    <  90°. 

Between  these  limits,  we  may  choose  various  values  for  a  and  ft, 
although  they  do  not  all  lead  to  an  equally  convenient  or  advan- 
tageous construction.  Fourneyron  makes  ft  =  90°,  and  a  =  30°  to 
33°,  some  constructors  make  ft  less,  and  others  greater  than  90°. 
When  ft  is  made  90°,  or  less,  the  curvature  of  the  buckets  is  greater 
than  when  ft  is  made  more  obtuse.  Great  curvature  involves  greater 
resistance  in  the  efflux,  and  hence  it  is  advisable  to  make  ft  rather 
obtuse  than  acute,  that  is,  to  make  ft  =  100°  to  110°.  The  angle 
o  must  then  be  50°  to  55°,  if  the  internal  pressure  is  to  balance  the 
external.  In  order,  however,  that  the  channels  formed  by  the 
guide-curves  may  not  diverge  too  much,  and  that  the  equilibrium  of 
pressure  may  not  be  disturbed  by  depressions  of  the  sluice,  this 
angle  is  made  from  30°  to  40°,  and  if  the  turbine  revolves  free  of 
back  water,  then  tt  should  not  be  more  than  25°  to  30°;  o  is,  how- 
ever, never  made  very  acute,  because  with  the  angle  a,  the  area  of 
the  orifices  of  discharge  varies,  and,  therefore,  the  quantity  of  water 
discharged  would  diminish,  or  the  diameter  of  the  wheel  must  be 
greater  for  a  given  quantity  of  water  expended.  On  the  other  hand, 
we  have  to  bear  in  mind,  that  the  losses  of  effect  increase  as  v2,  and 
that,  therefore,  cseteris  paribus,  a  turbine  will  yield  a  greater  effect 
—  will  be  more  efficient  —  when  revolving  slowly,  than  when  it  has 
a  great  velocity  of  rotation.  According  to  this,  the  construction 
should  be  so  arranged  that  a  and  ft  do  not  differ  widely  from  one 
another,  from  which  would  follow  an  internal  pressure  less  than  the 
external.  If  a  be  the  height  of  a  column  of  water  balancing  the 
atmospheric  pressure,  the  absolute  pressure  of  the  water  as  it  passes 
over  the  space  between  the  wheel  and  bottom  plate  is  a  +  a:,  and  if 
this  pressure  =  0,  then  the  water  flows  with  a  maximum  velocity 
c=  \/2<7  (hl  —  x)  =  v/2<7  (Ax  +  a)  from  the  reservoir.  If  a  +  x 
were  negative,  or  z  <  —  a,  there  would  arise  a  vacuum  at  the  point 
of  passage  of  the  water  into  the  wheel,  for  the  water  would  flow  even 
faster  through  the  wheel  than  it  flowed  on  to  it  from  the  reservoir, 
and  so  air  would  rush  in  from  the  exterior,  greatly  disturbing  the 
flow  of  the  water.  If,  therefore,  we  introduce  into  the  formula, 
instead  of 

*  =  A  ___  *  ___ 
1  +  cos.  2  a  —  cotff.  ft  sin.  2  a 
x  =  —  a,  we  then  have  : 

1  +  cos.  2  o  —  cotg.  ft  sin.  2  a  =  -  -  ,  hence: 


tana  ft 


h  +  a 
and,  therefore,  the  corresponding  best  velocity  of  rotation: 


TURBINES  WITHOUT  GUIDE-CURVES.  261 

9 


(h  +  a)sin.  2a  /  cos.  a > 2 (A  +  a) 
§  151.  Turbines  without  Guide-  Curves.  —  For  turbines  without 
guide-curves  we  may  make  a  =  90°,  because  the  water  flows  by  the 
shortest  way,  or  radially,  out  of  the  reservoir.  In  this  light  we 
have  to  consider  the  turbines  of  Combe^  Cadiat,  and  Whitelaw.  If 
we  introduce  into  the  formula  for  the  best  velocity  o  =  90°,  we  get: 


2  sin.  3  cos.  90°       „ 

cos.  8  \cos.  8 


-;  neglecting  prejudicial  resistances, 


however,  v1  =  1-^—  =  <»  .  But,  for  two  reasons,  the  wheel  can- 
not acquire  an  infinite  velocity.  There  is  a  limit,  in  the  first  place, 
when  the  mechanical  effect  at  disposition  is  absorbed  in  overcoming 
prejudicial  resistances,  that  is,  when 


h  =  [72  sin.  |  V  +  S  (^  tang.  3^  +  *  1 


and  in  the  second  place  when 

x  =•  —  a,  \.e.h  —  —  =  —  a,  or  —  =  a  +  A,  or, 

A  (r-L  .    _v*™-i*      V  =  a  +  A)  that  is,  when 
2g\r     sin.  (3  — 90°)} 

v  =  —cotg.  3  ^/2g  (a  +  A),  because  then  the  full  discharge  by  full 

* 


flow  ceases,  and  the  circumstances  are  quite  changed,  seeing  that 
the  water  cannot  flow  from  the  reservoir  in  quantities  sufficient  to 
supply  the  discharge  of  the  wheel  channels,  when  their  section  is 

If  we  introduce  into  the  above  formula: 

2gh 


the  experimental  co-efficients  £  and  *,  it  is  still  far  from  giving  us 
v  =  oo.  Now  for  the  most  accurate  construction  of  the  guide- 
curve  apparatus,  the  co-efficient  of  velocity  <f  is  not  greater  than 


262  INFLUENCE  OF  THE  POSITION  OF  THE  SLUICE. 

0,93,  and,  therefore,  the  co-efficient  of  resistance  £  corresponding 

=  — —  1  =  not  less  than ^ —  1  =  0,16,  or  about  16  per  cent. 

$  Ujc/  o 

In  the  case  of  turbines  without  this  apparatus,  this  resistance  does 
not  exist ;  but  still  there  remains  always  a  certain  loss  for  the  en- 
trance into  the  wheel  channels,  which,  in  Combe's  and  Cadiat's 
wheel,  does  not  probably  exceed  5  per  cent,  though  in  Whitelaw's  it 
may  be  taken  as  10  per  cent,  at  least,  for  the  channels  are  too  wide 
to  admit  of  the  supposition  that  the  whole  stream  of  water  has  the 
definite  direction  (ft).  The  co-efficient  *  corresponding  to  the  re- 
sistance from  the  curvature  and  friction  in  the  channels  may  be  set 
at  from  0,5  to  0,15,  as  we  shall  see  in  the  sequel,  and  hence,  for 
turbines  without  guide-curves,  we  have,  putting  *  =  0,1,  the  best 
velocity  , 

\  /  0,05  (tang,  ft)2  +  0,1  (JL)* 
and  for  Whitelaw's  reaction  wheel : 

«=,    r      ~^9^' 


0,1  (tang.  ft)2  +  0,1  t±-\ 


If,  again,  we  put  ft  =  60°,  and—  =  |,  we  have,  in  the  first  case: 
ri 


•  1,75 


),148  +  0,178 
and  in  the  second  case: 

_M =1,45 

),296  +  0,178 

For  other  reasons,  we  shall  hereafter  learn  that  the  most  advan- 
tageous velocity  of  rotation  is  not  even  equal  to  the  velocity  due  to 
the  height  h. 

In  order  that  the  water  may  enter  the  wheel  with  the  least  shock 
possible,  in  the  case  of  wheels  without  guide-curves,  the  equation 

-^  =— •  tang,  ft  must  be  satisfied.     But  as  F3  is  determined  by  the 

relative  position  of  the  sluice,  it  follows  that  the  maximum  effect  is 
got  for  a  certain  position  of  the  sluice. 

§  152.  Influence  of  the  Position  of  the  Sluice. — In  one  point  of 
view,  turbines  are  inferior  to  overshot  and  breast  wheels.  When  in 
these  wheels  there  is  only  a  small  quantity  of  water  available,  or  it 
is  only  required  to  produce  a  portion  of  the  power  of  the  fall,  and 
for  this  purpose  we  partially  close  the  sluice,  we  know  that  the 
efficiency  of  such  wheels  is  rather  increased  than  diminished  from 
the  cells  being  proportionally  less  filled.  In  the  turbine  the  con- 
trary is  the  case,  for  as  the  sluice  is  lowered,  the  water  enters  the 
wheel  under  circumstances  involving  greater  loss  of  effect.  This  is 


INFLUENCE  OF  THE  POSITION  OF  THE  SLUICE.  263 

a  circumstance  so  much  the  more  unfavorable,  inasmuch  as  it  is 
generally  requisite  to  economize  power  the  more,  as  the  water  supply 
fails.  The  loss,  however,  by  lowering  the  sluice,  is  never  very  great, 
as  the  following  investigation  proves. 

If  we  decompose  the  velocities  c  and  cl  into  their  radial  and  tan- 
gential components  c  sin.  a,  c  cos.  a,  ^  sin.  ft  and  ct  cos.  0,  and  sub- 
tract the  two  from  each  other,  there  remain  the  relative  velocities  : 

c  sin.  a  —  cl  sin.  J3,  and  c  cos.  a  —  cl  cos.  ft  ; 

as,  however,  the  water  has  the  velocity  vl  in  common  with  the  wheel, 
the  latter  relative  velocity  is  in  fact  =  c  cos.  o  —  c:  cos.  ft  —  vr  Ac- 
cording to  a  known  law,  the  loss  of  pressure  height  corresponding  to 
a  sudden  cessation  of  this  velocity  is  : 

y  =  —  [(c  sin.  a  —  ct  sin.  /3)2  +  (c  cos.  a  —  cl  cos.  ft  —  wx)s], 
or,  in  mechanical  effect  : 
T  =  y  Q  y  =  [(c  sin.  o  —  ct  sin.  0)2  +  (c  cos.  a  —  cx  cos.  ft  —  vj2]    y.. 

If  we  introduce  into  this  formula  c2  =  v  and  v1  =  -1  v,  further 

W  f 

c  =  —  ?  v  and  c1  =.  —  ?  v,  we  have,  as  the  loss  'of  mechanical  effect  : 
F  Fl 

F  sin.  «      F  sin 


From  this  we  may  judge  as  to  the  loss  of  effect  in  turbines  that 
do  not  fulfil  the  conditions  expressed  in  the  equation  : 

W  V      T 
Fi  sin.  a.  =  F  sin.  ft  and  JFl  cos.  a=  F  cos.  ft  -f  —  —  1  .  _i. 

However,  even  if  these  conditions  be  fulfilled  in  the  normal  state  of 
the  turbines'  working,  i.  e.,  when  the  sluice  is  fully  drawn,  they 
cannot  be  so  when  the  sluice  is  depressed,  and  F  becomes  Fx  .  The 
loss  of  mechanical  effect  then,  even  when  the  effect  is  a  maximum, 
viz.  :  c2  =  v,  is  : 

F2sin.a     J>TO.p\8      /F2cos.a     Fzcos.a 

"       ~~~  ~ 


or  substituting  F  sin.  ft  =  Fl  sin.  a  and  Fcos.  ft  +  —  -1  .  -1=  Fl  cos.  a, 

r-[ft: 


-  F 

If,  as  an  example,  we  put  ^-  =  |  h,  which  is  allowable  in  Four- 
neyron's  turbines,  we  have  : 


264  SLUICE  ADJUSTMENT. 

or,  for  the  sluice  half  open,  in  which  case : 


We  see  from  this,  that  this  loss  may  be  diminished  by  making  the 

ratios  — ?.  and  —  small,  that  is  to  say,  by  making  the  orifice  of  dis- 
F         r1 

charge  of  the  wheel  and  the  external  radius  small,  but  keeping  the 
orifices  and  radius  of  the  reservoir  large.     As  : 

F2  _       T-J  sin.  j3 

F  ~~  r  sin.  (p  —  a)' 
we  have  in  the  last  case  : 


and,  therefore,  for  js  =  90°,  and  a  =  40°,  Y=  0,57  Q  h  y,  or  there 
is  in  this  case  a  loss  of  57  per  cent,  of  the  effect. 

Generally,  when  the  sluice  is  much  depressed,  when  Fx  <  J  F, 
the  full  discharge  ceases,  that  is,  the  water  no  longer  fills  up  the 
wheel  channels,  the  wheel  becomes  a  pressure  turbine  only. 

§  153.  Sluice  Adjustment. — To  avoid,  or  at  least  to  diminish  the 
loss  of  mechanical  effect  which  results  from  lowering  the  sluice,  and 
in  order  to  retain  the  full  flow  of  water  through  the  wheel,  many 
devices  have  been  recently  introduced  by  Fourneyron  and  others. 
Fourneyron  divides  the  whole  depth  of  the  wheel  into  stages,  by 
introducing  horizontal  annular  division  plates,  dividing  the  total 
depth  into  two  or  three  separate  spaces,  so  that,  when  the  sluice  is 
lowered,  one  or  two  of  the  spaces  are  completely  shut  off,  and  the 
water  flows  through  the  other  subdivision.  This  arrangement  does 

Fig.  268. 


PRESSURE  TURBINES. 


265 


Fig. 


not  entirely  fulfil  its  object  ;  but  the  apparatus  shown  in  Fig.  268, 
invented  by  Combes,  does.  This  contrivance  consists  in  a  plate  or 
disc  DZ>,  between  the  two  shroudings  of  the  wheel,  which,  by  means 
of  rods  EE,  can  be  raised  or  depressed  by  means  of  a  simple  me- 
chanism, so  that  the  water  flowing  through  the  wheel  always  fills  up 
the  channels  open  to  it.  This  apparatus  fulfils  the  required  condi- 
tions, but  it  is  difficult,  and  very  costly  in  construction. 

The  turbines  of  Gallon,  and  also  those  of  Gentilhomme,  are  like- 
wise constructed  so  that  the  water  may  fill  up  the  channels,  how 
small  soever  the  quantity  supplied. 

Fig.  269  represents  a  part  of  Gallon's  wheel  in  elevation  and 
section.  This  shows  that  the  guide-curves 
are  covered  in  on  top,  and  in  the  inside  by 
sluices  E,  E  .  .  .  ,  each  of  which  closes  two 
apertures.  To  regulate  the  discharge  of 
water  by  this  arrangement,  it  is  only  neces- 
sary to  keep  a  certain  number  of  apertures 
closed.  Although  this  arrangement  cer- 
tainly provides  against  impact  on  the  wheel, 
yet  it  is  imperfect,  inasmuch  as  the  water 
can  work  little,  if  at  all,  by  reaction,  as  it 
does  not  run  through  the  wheel  channels  in 
an  unbroken  stream.  In  this  alternate  fill- 
ing  and  emptying  of  the  wheel-channels, 
the  velocities  c,  cl  and  c2  undergo  continual 
variations  unless  x  =  0,  that  is  j3  =  2  a. 
If,  for  example,  the  wheel  -channels  not 
being  filled,  c  =  i/Zgh^  we  should  have 
c=  \/2g-  (h  —  x),  when  the  water  stream 
filled  up  the  channels.  Thus  for  each  fill- 
ing and  emptying,  or  while  the  wheel  passes 
from  one  open  aperture  to  another,  the  velo- 
city c  continually  oscillates  between  the  limits 
</2gh,  and  \/2g  (h  —  x).  As  the  maximum  effect  can  only  be  ob- 
tained tor  determinate  values  of  v  and  c2  =  —  ,  it  is  quite  evident 

^2 

that,  as  c2  varies,  we  fail  in  this. 

For  Gentilhomme's  turbine,  the  same  object  is  attained  by  circu- 
lar sectors,  which  are  so  placed  by  means  of  mechanism,  that  they 
close  a  part  of  the  guide-curve  apparatus;  an  arrangement  evidently 
even  more  imperfect  than  Gallon's.  Hanel,  a  German  engineer, 
describes  an  arrangement  of  sluice  for  effecting  the  objects  now  in 
question,  very  similar  to  that  of  Combe's.  (See  "  Deutsche  Gewerb- 
zeitung,  1846.) 

§  154.  Pressure  Turbines.—  This  is  the  place  to  compare  the 

reaction  turbines  hitherto  under  discussion  with  the  impact  and  pres- 

sure turbines,  into  which  they  always  become  converted  when  the 

sluice  (7,  Fig.  270,  closes  the  greater  part  of  the  depth  of  the  wheel 

VOL.  II.—  23 


266  PRESSURE  TURBINES. 

AS.     As  the  water  W  only  partially  fills  the  section  of  the  wheel 
channels,  the  remainder   is  filled   with   air, 
Fig.  270.  unless  the  wheel  works  free  of  back  water, 

and,  therefore,  the  pressure  on  the  outside  of 
the  wheel  is  that  of  the  atmosphere;  the  velo- 
city is  c  =  \/2gh,  and  is  independent  of  the 
motion  of  the  wheel.  But  for  the  velocity  of 
discharge  we  have  c22=  2gh  +  v2 — 2  c  v^  cos.  o, 
and  for  the  maximum  effect  c2  =  v,  and  then 
for  these  wheels,  the  expression 

2  c  v.  cos.  a  =  2gh,  or  substituting  c  =  \/2#A,  vl  =       ~&    ,  obtains. 

—  COS.  a 

In  the  case  of  reaction  wheels,  we  found : 

v l  =  \/gh  (1  —  tang.  a.  cotg.  0); 
and  hence  we  perceive  that  the  conditions  of  maximum  effect  are 

identical  in  both  cases,  if =1  —  tang,  a  cotg.  j3,  or  if  tang. 

L.  COS.  a 

0  =  tang.  2  o,  that  is  ]3  =  2  a;  which  result  we  have  already  ascer- 
tained in  the  form  of  the  condition  that  x  =  0.  There  is,  therefore,  an 
essential  difference  in  the  turbines  of  the  two  classes,  in  as  far  as 
the  velocity  for  the  maximum  of  effect  does  not  in  the  one  depend 
upon  j3,  whilst  it  does  in  the  other;  and  it  is  only  when  P  =  2  a  that 
this  velocity  is  the  same  for  both.  While,  therefore,  for  reaction 
wheels  the  velocity  v1  may  be  made  to  vary  within  wide  limits  by 
selection  of  the  angle  j3,  we  have  no  such  joice  in  the  case  of  impact 
turbines. 

In  reference  to  the  effect  of  both  wheels,  we  adduce  the  following 
facts.  When,  in  a  reaction  wheel,  the  sluice  is  gradually  lowered, 
the  efficiency  diminishes,  and  when  it  is  so  far  lowered  that  the 
water  does  not  fill  up  the  wheel  channels,  the  turbine  then  passing 
into  a  pressure-turbine,  the  efficiency  suddenly  rises,  because  the 
loss  of  mechanical  effect  by  the  sudden  change  of  velocity  ceases. 
By  lowering  the  sluice  still  further,  the  further  loss  of  effect  is  in- 
considerable. According  to  this,  the  pressure  turbine  seems  to  be 
a  better  wheel  than  the  reaction  turbine ;  but  from  other  circum- 
stances the  advantages  do  not  preponderate,  and  can  only  be  ac- 
corded when  the  supply  of  water  is  liable  to  much  fluctuation. 

As  the  water  entering  the  wheel  finds  a  much  greater  sectional 
area  than  at  its  velocity  it  can  fill,  it  gets  into  an  irregular  oscilla- 
tory motion,  and  not  only  does  not  discharge  with  the  velocity  c2  above 
calculated,  but  also  loses  a  part  of  its  vis  viva  absorbed  in  creating  the 
eddying  irregular  motion  alluded  to  (and  which  no  doubt  isconsumed  in 
raising  the  temperature  of  the  water,  TR.).  Numerous  experiments 
have  proved  this,  and  these  may  be  repeated  with  any  turbine,  if  it 
be  made  to  revolve  with  the  best  velocity,  first  as  a  reaction  wheel, 
and  then  as  a  pressure  wheel.  Turbines  always  give  a  greater 
effect  for  an  open  sluice  and  full  discharge  than  when  the  sluice  is 


MECHANICAL  EFFECT  OF  TURBINES.  267 

lowered  and  the  water  does  not  fill  the  section   of  the  wheel's 
channels. 

When  turbines  work  under  water,  the  flow  is  always  full  through 
them,  and  these  wheels  are,  therefore,  always  reaction  wheels.  A 
greater  efficiency  is  naturally  to  be  expected  from  these  when  the 
sluice  is  fully  opened,  than  from  the  turbines  revolving  free  of  back 
water;  on  the  other  hand,  we  may  safely  assume  that,  when  the  sluice 
is  lowered  so  that  only  f  or  less  of  the  depth  of  wheel  is  open,  the 
efficiency  of  the  reaction  wheel  will  be  less  than  that  of  the  pressure 
turbine.  From  this  we  can  easily  understand  the  great  advantage 
of  introducing  the  horizontal  dividing  plates. 

Remark.  All  the  older  turbines  of  Fourneyron  were  pressure  turbines  ;  but,  as  experience 
pointed  out  the  greater  efficiency  of  the  reaction  turbines,  almost  all  turbines  are  now 
reduced  to  this  principle. 

§  155.  Mechanical  Effect  of  Turbines.  —  We  can  now  calculate 
the  mechanical  effect  of  turbines.  The  effect,  which  is  not  taken 
from  the  water  if  it  flow  from  the  wheel  with  an  absolute  velocity  : 

w  =  y/c£  +  v2  —  2  ca  v  cos.  8,  or  if  cs  =  »,#;  =  2  v  sin.  -, 
4  v2    sin.  - 


2g  2g 

f    The  effect  which  the  water  loses  in  the  guide-curve  apparatus,  or  in 


getting  into  the  wheel,  is  : 


in  which  f  =  0,10  to  0,20,  according  as  the  wheel  is  provided  with 
guide-curves  or  not. 

The  third  loss  of  effect  is  =  *  .  £?-  Q  y,  and  consists  of  the  friction 

and  curve  resistance.  The  resistance  in  passing  round  the  curve 
may  be  found  by  the  rules  in  Vol.  I.  §  334.  The  corresponding 

loss  of  head  =  C,  -  -  —  -  {—\  in  which  £  is  a  co-efficient  dependent 
A    2g    \FJ 

on  the  ratio  —  of  the  half  of  the  mean  width  of  the  channel  to  the 

2a 

mean  radius,  t  the  central  angle,  F  the  sum  of  the  mean  section  of 
the  channels.  If  n  be  the  number  of  channels  or  of  buckets,  and  if 
e  be  the  mean  height  of  a  channel,  then  F=nde,  and,  therefore, 
the  height  of  head  lost  by  the  resistance  in  the  curves  : 


t/=  ^  *  .    L  .    --,  or  if  we  put: 
1  *     2g     \nde/ 

* 


=  0,124  +  3,104  (—}*  (as  in  Vol.  I.  §  334),  then  : 
\2#/ 


268  MECHANICAL  EFFECT  OF  TURBINES. 

and  the  loss  of  mechanical  effect  corresponding,  is  : 

24  +  3,104  (— 

\2a 

or  putting  Q*  =  (F2  1>)2, 


The  wheel  buckets  consist  usually  of  two  parts  of  different  curva- 
tures, and,  hence,  L3  would  be  made  up  of  two  items.  It  is  evident 
from  the  above,  that  this  source  of  resistance  increases  the  wider 
the  channels  are,  and  the  less  the  radius  of  curvature  a.  Hence  it 
is  advisable  to  make  ^  obtuse,  so  as  to  diminish  the  curvature  of  the 
buckets,  which  is  also  advisable  in  respect  of  a  full  flow  through 
them. 

The  resistance  from  friction  is  to  be  calculated  according  to  Vol. 
I.  §  330.  If  £2  be  the  co-efficient  of  friction,  p  the  mean  periphery, 
and  I  the  length  of  the  wheel  channel,  the  height  due  to  the  resist- 
ance from  friction  : 


2'  de  '  2g  '  \^d~e     ""d~e  '  \^~d~e     '  2g 
and  the  loss  of  effect  corresponding  : 


_ 

or,  if  p  =  2  (d  +  e),  and  £,  =  0,0144  +  0,0169     \nde  be  intro- 
duced : 

i4-  (0,0144  +  0,0169   ]**•)  .  y+/>l  .  (  JLV  .  ^  Q  r. 
\    Q  /        2de        \n  dej      2g 

If,  lastly,  Cr  be  the  weight  of  the  turbine  in  revolution,  and  r2 
the  radius  of  the  pivot,  the  loss  of  effect  by  the  friction,  then,  is  : 

l.  L  §171). 


If,  now,  we  deduct  these  five  losses  of  effect  from  the  power  at 
disposition,  there  remains  of  useful  effect: 


In  order  to  have  this  mechanical  effect  great,  it  is  necessary  to 
make  the  velocity  of  rotation  v,  the  area  of  the  orifice  F2,  the  orifice 
angle  S  small.  In  how  far  this  is  possible  we  have  above  shown. 

It  is  only  in  the  case  of  turbines  working  under  water  that  the 
height  h  is  to  be  measured  from  water  surface  to  water  surface. 
For  turbines  working  in  air,  h  is  to  be  measured  from  the  upper 
surface  to  the  centre  of  the  wheel.  In  the  latter  case,  the  freeing 
the  wheel  of  back  water  involves  a  loss  of  head,  measured  by  the 
distance  from  the  centre  of  the  wheel  to  the  surface  of  the  race, 


CONSTRUCTION  OF  GUIDE-CURVE  TURBINES.  269 

whilst  for  wheels  working  under  water  there  is  a  loss  from  the  resist- 
ance of  the  medium. 

Remark.  For  high  pressure  turbines  there  is  an  additional  source  of  loss  in  the  resist- 
ance of  the  flow  of  water  through  the  pressure  pipe. 

§  156.  Construction  of  Guide-Curve  Turbines.  —  Let  us  now  en- 
deavor to  deduce  the  rules  necessary  for  planning  a  wheel  consist- 
ently with  the  ahove  principles.  We  may  of  course  assume  the 
quantity  of  water  discharged  Q,  and  the  fall  h  to  be  given  ;  and  if, 
instead  of  Q,  the  useful  effect  L  were  given,  we  might  then  at  least 
derive  Q  from  L,  and  the  efficiency  n  (about  0,75)  by  the  formula: 

Q  =  —  -  —     The  remaining  quantities  r,  r:  o,  0,  8,  »,  n,  e,  &c.,  are 

1  «-  7 

determined,  partly  by  discretion,  partly  by  experience,  and  partly 
by  theory. 

The  angle  a  is  generally  assumed.  For  wheels  without  guide- 
curves  it  is  taken  as  90°,  but  for  wheels  with  guide-curves  it  must 
be  made  from  25°  to  40°;  the  former  for  high  falls,  the  latter  for 
small  falls,  in  order  that,  in  the  former  case,  the  orifices  may  not 
be  too  large,  and  in  the  latter  not  too  small,  or,  in  order  that,  in  the 
former  case,  the  wheels  may  not  be  too  small  in  diameter,  and  in  the 
latter  not  too  great.  The  angle  j3  is,  in  a  certain  degree,  fixed  by 
the  value  of  o.  That  the  water  may  enter  the  wheel  without  pres- 
sure on  the  free  space,  we  must  have  £  =  2  a  ;  but  as  this  pressure 
diminishes  as  the  sluice  is  depressed,  in  order  to  prevent  negative 
pressure,  j3  is  made  greater  than  2  o,  and  probably  /3  =  2  o  +  30° 
to  2  o  -}-  50°  are  good  limits;  the  former  in  high  falls,  the  latter  in 
low  falls. 

The  ratio  v  =  —  of  the  internal  and  external  radii  of  the  wheel 

rx 

falls  within  the  limits  of  1,25  and  1,5.  For  reasons  easily  under- 
stood, the  smaller  ratio  is  to  be  chosen  for  large  values  of  /3,  and  for 
wheels  of  considerable  diameter,  and  vice  versa. 

In  order  further  to  determine  the  radius  of  the  wheel,  and  of  the 
reservoir,  we  shall,  as  is  the  case  in  the  best  turbines  hitherto  made, 
require  fulfillment  of  the  condition  that  the  velocity  of  the  water 
in  the  reservoir  shall  not  exceed  3  feet  per  second.  If  we  adopt 
this  velocity  as  ground  work  of  our  calculation,  and  leave  out  of  the 
question  the  section  of  the  upright  pipe  encasing  the  axle,  and  that 
of  the  sluice,  then  Q  =  3  *  rf,  and,  therefore,  inversely,  the  radius 
of  the  reservoir,  or  the  internal  radius  of  the  wheel  : 

r1==     !_?_  =  0,326  -v/Qj  when  rt  is  in  feet,  and  Q  in  cubic  feet. 

\  3  K 

From  this  radius  we  get  the  external  radius  r  =  v  rt.  Ine  velocity 
at  the  inner  periphery  of  the  wheel  is  determined  by  the  formula  : 


pcQ*..       r(     sin.p      \*      .  /r_\ 
/!  —  •)    ^        Un.03—  a)/  Vf,/ 


«n. 

23* 


270 


CONSTRUCTION  OF  GUIDE-CURVE  TURBINES. 


into  which  we  must,  in  the  first  place,  introduce  an  approximate 
value  of  x.     From  this,  however,  we  get  the  velocity  of  discharge  : 

v,  sin.  3          j  ,,  -r,       O       Q  sin.  (3  —  o)      »    ,  , 

c  =  —  l-  --  —  ,  and  the  section  F  ==  —  =  _  --  £  --  '-  ;   further. 
sin.  ()3  —  a)  c  v1  sin.  s 

i         i     .,       /.  c  sin.  a          v,  sin.  a  ,     , 

the  velocity  of  entrance  c.  =  -  =  —  l  --     and  the   sec- 


sin.  /3 


tion  F.  =  .    = 


—  --  , 
sin.  (fl  —  o) 

the  Yeiocity  Of  the  external 


Ffe.  271. 


periphery  of  the  wheel,  and  of  the  exit  from  it:  v  =  c2  =  — «„  and 
the  contents  of  the  united  orifices  of  discharge  from  the  wheel : 

jF,  =  —  =  ^i  .  —  =  _L  .  —     Besides  this,  we  ascertain  the  number 
c2       r     vl       r      vl 

of  revolutions  of  the  wheel  per 

•          *      V  30  V  0    rr  V 

mm.  to  be :  u  =  =  9,55  -. 

n r  r 

In  order  to  find  the  height 
of  the  wheel,  or  of  the  orifice  e, 
we  pursue  the  following  method. 
If  nl  be  the  number  of  guide- 
curves,  and  d1  the  least  distance 
AN  (Fig.  271),  between  any 
two  of  the  guide-curves  at  the 
entrance  on  to  the  wheel,  then 
nldle  =  F.  If,  further,  dl  and 
e  be  in  a  determinate  proportion 


to  each  other  : 


—  ,  then 


d1  =  AAl  sin.  o 


and,  hence, 


nl 4j  d*  =  F;  and  if  «  be  the 
thickness  of  one  of  the  guide- 
curves,  we  may  put  with  tolera- 
ble accuracy : 
2  *  rl  sin,  o g 

ni 


so  that,  by  inversion,  the  number  of  buckets  required  : 
4,  (2  tt  r.  sin.  a  —  n.  «,)2 

"-       /  j.    a~ ' 

or,  approximately  :  =  **  ^    *—1  8m'  *'  ,  for  which  a  whole  number 

would  be  taken.     If  n1  be  once  fixed,  then  : 
,  ^_  2  *  r,  sin. a          _  & 


r  sn.  a.  —  n 


and  the  height  of  wheel :  e 


sin.  a  —  nls1 


CONSTRUCTION  OF  THE  BUCKET.  271 

§  157.  We  have  still  to  deduce  rules  by  which  to  calculate  the 
number  of  wheel-buckets,  and  the  dimensions  of  the  orifices  of  the 
wheel.  The  orifices  of  discharge,  the  united  area  of  which  is 

Fz  =  ^,  is  not  the  outer  periphery  of  the  wheel,  but  the  section 

BJ),  BJ)V  &c.,  through  the  outer  end  of  the  buckets  Bv  Bv  &c. 
(Fig.  271).  Again,  for  r  in  the  above  formulas  we  are  not  to  under- 
stand the  radius  of  the  outer  periphery,  but  the  distance  CM  of  the 
centre  of  the  orifice  B^D  from  the  axis  of  rotation,  and,  in  like 
manner,  v  is  not  the  velocity  of  rotation  of  B,  but  of  M.  If,  now, 
a  be  the  angle  SMT,  which  the  axis  of  the  stream  flowing  through 
BD  makes  with  the  tangent  MT,  or  the  normal  to  the  radius 
CM=  r;  and,  further,  if  n  be  the  number  of  wheel-buckets,  *  their 
thickness,  d  the  width  BVD  of  the  orifices  of  discharge,  and  ^  the 

ratio  -,  we  may  put  :nde  =  n$d?= =  F2,  therefore,  inversely, 

d  4- 

•p 

the  number  of  wheel-buckets  n  =  —/•     Again,  as 

2  *  r  sin.  8  —  n  8  =  n  d  =  —  =  ?*. 
4         e 

we  have  for  the  angle  of  discharge  sin.  «  =  -P,  (*  +  *«). 

2  n  r  & 

This  angle  &  should  not  in  any  case  be  more  than  20°,  and,  there- 
fore, if  it  comes  out  more  than  this,  by  the  latter  formula,  some  one 
or  more  of  the  elements  composing  it  must  be  changed.  Thus,  for 

example,  for  this  purpose  F2  =  !l  .  -^  may  be  made  less,  f.  e.,  v,,  or 

T        Vj 

which  amounts  to  the  same,  the  difference  between  a  and  p  may  be 
made  greater.  Some  engineers  have  endeavored  to  keep  6  small, 
by  making  the  wheel  deeper  at  the  outside  than  in  the  inside,  giving 
two  values  to  e  (Fig.  269).  This,  however,  has  the  disadvantage, 
that  full  discharge  is  thereby  interfered  with — at  least  in  wheels 
revolving  free  of  back  water,  and  that  the  water  follows  the  wheel 
shroudings  when  these  diverge  from  each  other  to  any  considerable 

extent.     As  to  the  ratio  4  =  -,  its  influence  on  e  and  «  is  but  trifling. 

d 

It  is  within  the  limits  2  and  5  in  wheels  giving  good  results.  The 
small  value  refers  to  small  wheels,  and  viceversd;  for,  otherwise,  the 
channels  fall  out  too  wide,  and  the  full  flow  is  liable  to  be  lost. 

§  158.  Construction  of  the  Bucket.— The  buckets  are  generally 
circular  arcs.  For  the  guide-curves  one  arc  is  sufficient ;  but  for 
the  wheel-curves,  or  buckets,  two  arcs,  tangential  to  each  other,  are 
usually  required.  How  to  fix  the  radius  of  these  arcs,  and  how-to 
combine  them,  may  be  explained  as  follows:  With  CM  =  r,  Fig.  272, 
describe  a  circle,  draw  the  tangent  MT,  and  upon  it  set  off  the  angle 
of  discharge  SMT=  J,  as  determined  above.  Draw  M 0  at  right 
angles  to  MS,  and  set  off  on  each  side  of  M,  MD  =•  MBl  =  £  rf. 


272 


CONSTRUCTION  OF  THE  BUCKET. 


Now  draw  the  radius  CBV  and  from  C  lay  off  the  angle  B,CB  =  <j>°, 

as  determined  by  the  formula 


Fig.  272. 


S  ATP 

.         Also,  from 
r  sin.  S 

centre,    draw    circles 
The  first 


n 

C,    as 

through  B1  and  D. 
of  these  circles  gives  the  exter- 
nal periphery  of  the  wheel,  and 
the  points  -B,  B,,  &c.,  are  the 
outer  ends  of  the  buckets.  If 
we  draw  J30,  so  that  BOD  = 
BCBl  =  t,  we  have  in  0  the 
centre,  and  in  BO  =  DO,  the 
radius  of  the  arc  DB  forming 
the  outer  portion  of  the  bucket. 
If  we  make  B,0,  =  DO,  we 
have  the  centre  01  of  the  outer 
piece  BlDl  of  the  next  follow- 
ing bucket,  &c.  &c.  The  radius 
OB  =  OD  =  a  of  the  arc  BD 
may  be  also  determined  by  solution  of  the  triangle  MOM,.  We 

have  :   —  =  sin'  MMi°,  but  the  cord  MM,  =  2r  sin.  * 
MM,      sin.  MOM,  2 


MOM,  = 


and  MM,0 


90  +  6  —  ~-j  therefore, 


OM 


sn.  * 


and  the  radius  a  required 


r  cos.  is ^- 


By  this  method  of  construction,  the  end  B,  of  the  bucket  is  quite 
parallel  to  the  element  D  opposite,  and,  therefore,  the  stream  flows 
out  without  contraction.  If  this  parallelism  be  not  effected,  it  is 
always  disadvantageous;  if  the  tangents  to  B  and  D  diverge  out- 
wards, there  is  danger  of  losing  the  full  flotv,  and  if  they  converge, 
there  arises  a  partial  contraction,  and  the  stream  then  strikes  upon 
the  outer  surface  of  BD  (Vol.  I.  §  319). 

The  inner  piece  DA  of  a  wheel-bucket  may  generally  be  formed 
of  one  arc  of  a  circle.  The  radius  KD  =  KA  =  a,  of  this  circle  is 
found  as  follows :  In  the  triangle 

CMK,  CM=  r,  MK=  a,  +  fj,  and  /  CMK  =  SMT=  a, 


CONSTRUCTION  OF  THE  BUCKET.  273 


—  —  2ral  cos.  S  —  rd  cos.  8  =  r,2  -f  2  r  a  cos. 


In  the  triangle  CAK,  on  the^ther  hand,  CA  =  rv  AK  =  aw  and 
CAK  =  180°  — 0,  therefore,  C#2  =  r,2  +  a,2  +  2  r:  a,  cos.  0.     By 
equating  the  two  expressions,  we  have : 
d2 
T 
and  hence  the  radius  required: 

d2 

u  = 4^ 

2  (r  cos.  6  +  rx  cos.  /3)  —  d 

As  to  the  arc  to  be  adopted  as  the  curvature  of  the  guide-curves, 
we  get  its  radius  and  centre  by  drawing  AL  at  the  known  angle 
o  to  the  tangent  AH  of  the  inner  circumference  of  the  wheel.  Raise 
a  perpendicular  AG  to  it,  and  cut  this  in  G  by  another  normal, 
raised  from  the  middle  point  E  of  the  radius  CA.  This  point  G 
is  the  centre  of  the  guide-curve  AF,  which  may  be  drawn  either 
quite  up  to  the  case  pipe  of  the  axle,  or  to  within  any  convenient 
distance  of  it.  The  radius  GA  =  GC  =  as  of  this  bucket  is: 


2      2  cos.  a 

The  centres  of  the  arcs  forming  the  outer  arcs  are  in  circles  de- 
scribed with  the  radii  CO,  CK,  and  CGr. 

Example.  It  is  required  to  determine  all  the  proportions  and  lines  of  construction  of  a 
Fourneyron's  turbine  for  a  fall  of  5  feet,  with  30  cubic  feet  of  water  per  second.     We 

shall  take  a.  =  30°,  and  B  =  110°,  and  adopt  the  ratio  -  =»  =  1,35.  This  being  as- 
sumed, we  have,  from  the  rules  above  given,  the  internal  radius  rt  =  0,326  ^Q  =  1,785 
feet,  for  which  we  take  1,8  feet  Hence  r,  =  CM  (Fig.  272),  the  external  radius 
=  1,8  X  1,35  =  2,43  feet,  for  which  we  shall  put  2,45.  The  width  of  the  shrouding, 
therefore,  measured  to  the  centre  of  the  orifice  of  discharge,  =  2,4 5 —  1,8  =  0,65  feet. 
Neglecting  prejudicial  resistance,  the  best  velocity  of  the  wheel  is: 

v ,  =  v/gA  (1  —  tang.  a.  cotg.  8)  =  v/5 .  3 1 ,2 5  ( 1  +  tang.  30°  cotg.  70°) 

=  ^156,25  .  1,21014  =  13,75  feet, 
but,  taking  these  resistances  into  account,  if  f^  0,1 8  and  *s=0,06: 


>/ 1,6527  +  0,1 639 +  0,1 176       VJ.9342 
and  then  »  =  &,  =  »»,=  1,35  .  12,71  =  17,15  feet.     The  velocity  of  discharge 

c  =  _^'_*_  =  12'71  ""•  70°  =  12,13  feet. 


tin.  (6  —  *)  tin.  80° 

r  of  revolutions  per  minute  i 
From  this  we  have  the  areas  of  the  orifices  of  discharge: 


The  number  of  revolutions  per  minute  is  u  =  9,55  .  Ef  =  —  -  '—  — 


274  CONSTRUCTION  OF  THE  BUCKET. 

F  =  ?  =    3°     =  2,473  square  feet,  and  F~  =  ®  =  ?  =    3°     =  1,748  square  feet. 

e        12,15  c2       v         17,15 

Assuming  the  thickness  of  the  buckets  to  be  2j  lines  =  0,017  feet,  and  supposing  tha  t 

the  ratio  of  the  depth  to  the  width  of  the  orifices  -|,  =  —  =  |,  we  have  as  the  requisite 

dl 
number  of  buckets  : 

w  _4.  (2«-r,«n.tt  —  n^)'  _  3  .  (5,655  —  0,017  n,)»  _  ^ 

~F  2,473 

and,  hence,  we  have  as  the  height  of  the  wheel,  or  of  the  orifices  : 

2'473 


__  0,4808  fee, 

2wr,  tin.  a  —  n,«,        5,655  —  0,51         5,145 
Supposing  ^  =  £  ,  the  number  of  buckets  would  be: 

n  =  i*»  =  5  '  1'748  =    8'74    =  37,8,  for  which  we  may  adopt  36.     From  this  we 

e*  0,4808'         0,2311 

get  the  required  angle  of  discharge: 

tin  j_-F»(e+^*)   _  I>748  (0,4808+0,017  .  5) 

2wret  2  -a  .  2,45  .  0,4  80S1 

_    1,748.0,5658 


4,9  .  v  .  0,4808' 


oa780  consequently  ,_  16o   g/  and 


d  =    •  =  =  0,1010  feet. 

ne       36.0,4808 
If  the  turbine  is  to  be  clear  of  the  back  water,  it  must  be  raised  a  certain  height  above 

the  surface  of  the  tail-race  ;  and  as  the  half  height  of  the  wheel  —  =  0,2404   feet,   this 

distance  may  be  estimated  at  0,5  feet.  If  this  excess  of  fall  does  not  exist,  then  the  cal- 
culations must  be  based  on  a  fall  of  4  £  feet,  instead  of  on  that  of  5  feet.  In  order  to 
judge  of  the  loss  of  water,  we  have  to  find  the  amount  of  x,  the  excess  of  pressure  of  the 
water  passing  under  the  sluice.  We  have  : 

x=h  —  (1+f)  fL  =  5—  1,18  .  0,016.  12,13'=  5  —  2,778  =  2,222  feet,  and  the  velo- 

city corresponding  =  7,906  ^2,222  =  1  1,78  feet.  If,  therefore,  the  space  between  the 
wheel  and  the  bottom  plate  be  ^  inch,  its  area  is  :  2  .  1,8  .  it  .  ^  *L  =  0,0393  square 

feet,  and,  therefore,  the  quantity  of  water  escaping:  Q,  =  11,78  .0,0393  =  0,46  cubic 
feet.  To  diminish  this  loss,  which  will  be  the  less  the  lower  the  sluice,  the  filtingup  of 
the  wheel  must  be  very  accurately  done,  so  that  the  space  between  the  wheel  and  bot- 
tom plate  may  be  as  small  as  possible;  or,  by  increasing  c  and  making  $  less,  x  must  be 
reduced  as  low  as  possible. 

360° 

The  dividing  angle  of  the  wheel  is  _  =  10°;   but    the   thickness  of  the  buckets 
36 

takes  up  an  angle  =     *      =  _  2?2iZ  _  =  0,02497,  or  an  angle  =  1°,  26',  hence 

rsin.S       2,45  sin.  16°,  8' 

4.  =  8°,  34',  the  angle  of  curvature  of  one  part  of  the  bucket  The  radius  correspond- 
ing is  : 


_  ;co..L_  = 

°       ' 


cos.  i  ?  cos.  4°,  17' 

The  radius  of  the  second  part  of  the  wheel-bucket: 
a-  =  >"-r.«-rrfcB..»+»tf  =  2,785-0,2377^2,5473  ,,  f^ 

2  (rcos  fr-f-r.cos.g)—  d          3,476—0,101  3,375 

The  corresponding  angle  of  curvature  is  :  <f,  =  180°  —  8  —  >+  v  —  T,  in  which 

tang,  a-  =      a«  nn-  &      anj  tang.r=      °l  nn'  *     ;  expressed  numerically: 
r,  —  a,  cos.  B  r  —  a,  cos.  J 

^,  =  70°—  16°,  8'  +  24°,  42'  —  6°,  56'  =  71°,  38'. 

These  investigations  afford  us  the  necessary  elements  for  the  construction  of  a 
turbine  for  the  fall  in  question,  and  we  have  now  only  to  calculate  the  useful  effect 
that  such  a  machine  will  yield.  The  absolute  velocity  of  the  water  discharged  is 
ic  =  2  r,  sin.  $  >  =  2  .  17,15  sin.  8°,  4'  =  4,813  feet,  and,  hence,  the  loss  of  fall  correspond- 

ing =—=  0,016  .4,  8  13'=  0,371  feet.     Again,  the  loss  of  fall  occasioned  by  the  resist- 
9g 


TURBINES  WITHOUT  GUIDE-CURVES.  275 

ance  in  the  guide-curves  =  0,lS  |L=  0,18  .  0,16  .  12,13'  =  0,423  feet.     The  loss   of 

fall  arising  from  the  hydraulic  resistances,  maybe  estimated  as  follows:  From  an  accu- 
rate  drawing  of  the  wheel,  and  the  results  of  the  calculations  given  above,  it  will  be 
found  that  each  wheel  channel  consists  of  two  parts,  of  which  the  one  is  0,11  feet  wide, 
and  0,2  long;  the  radius  2,35  feet, the  central  angle  4$°,  and  the  other  is  0,21  feet  wide, 
and  0,95  feet  long,  the  radius  of  curvature  0,755  feet,  and  central  angle  of  71°,  38'. 
From  this  we  deduce  the  co-efficient  of  resistance  for  the  smaller  part : 
£,  =  0,124  +  3,104  QL)£=  0,124+  3.104YMI)*  =0,124 

and  for  the  larger:  £,  =0,124+  3,104  .  (^21)1=0,127.     Again,  the  angle  ratio  for 

V  1,51  / 

the  first  part  is  1  =  M  =  0,025,  and  for  the  second  1L5?  =  0,398.     Again,  the  sec- 
tion of  the  first  is  _£•_  =  36    Qi1;74^uAo  =  0,91 8,  and  for  the  second  part 


=  — — TT-2- — -          =  0,481,  and  hence  we  have  for  the  co-efficient  of  the  whole  resist- 
ance arising  from  curvature : 

«,  =  0,124  .  0,025  .  0,9182 -f  0.127  .  Q.398  .  0,48P=  0,0026  +  0,0117  =  0,0143. 
Further,  the  co-efficient  of  friction  in  the  first  part : 

£2  =  0,0144  +0,0169     /llf  =  0,0144  +  0,0169    I36'0'11  •  °'4808 
W    Q  >/        30 

0,0187,  and  for  the  second  =  0,0144  +  0,0169     I36  -0.21  -0.4808  _  Qi0203       ^ 

l  for  the  first  part  =  = 1'117' and  for  the  second 


;  3,250,  and  from  this  we  have  the  co  efficient  of  friction  for  the 

whole  channel : 

x,  =  0,0187.  1,117  .  0,91 8'+ 0,0203  .  3,250  .  0,4 81*  =  0,0 176  + 0,0152  =0,0328, 
and  hence,  lastly,  the  co-efficient  for  all  the  resistances  in  a  wheel-channel  is: 
*  =  *,  +  *,  =  0,0143  +  0,0328  =  0.0471,  and  the  loss  of  fall  corresponding   to  this 

=  x  .  ^=0,0471  .  0,016  .  17,15*  =  0,222  feet.  The  three  losses  of  fall  just  esti- 
mated =  0,37 1  +  0.424  +  0,222  =1.017  feet,  and  thus  there  remains  of  the  total  effect 
at  disposition,  Q  hg  =  30  .  5  .  66  =  9900  feet  Ibs.,  only 

P»  =  30  .  (5 —  1,017)  66  =  7886  Ibs.  as  useful  effect.  There  is,  however,  some  por- 
tion of  this  consumed  by  the  friction  of  the  pivot.  If  the  weight  of  the  wheel,  &c., 
be  2000  Ibs.,  and  supposing  the  radius  of  the  pivot  =  l£  inch,  and  the  co-efficient 
of  friction  0,075,  the  mechanical  effect  consumed  by  the  friction  of  the  pivot 

=  !*/  G»,  =       *      .  0,075  .  2000  .  12,71  =  132  feet  Ibs.    (Suppose  the  co  efficient  of 

ri  8.1,8 

friction  0,12,  which  is  more  likely,  then  the  friction  =  210  feet  Ibs.)  The  useful  effect 
available,  directly  at  the  axle  of  the  wheel,  is  then  : 

Z  =  7886  —  210  =7676  feet  Ibs.  =  13,9  horse  power.  If  we  assume  0,5  feet  lost 
besides,  by  keeping  the  wheel  free  of  the  water  in  the  race,  then  the  efficiency : 

,  =     L    =         7676        =  0,705." 
Q  h  y       30  .  5,5  .  66 

§  159.  Turbines  without  Guide-Curves. — The  proportions  of  tur- 
bines without  guide-curves  are  only  partly  deducible  in  the  manner 
of  turbines  with  guide-curves.  The  angle  o  is  in  these  90°,  and  the 

*  [It  will  be  observed  that,  in  working  this  example,  we  have  retained  the  co-efficients 
applicable  to  the  Prussian  weights  and  measures,  viz.:  _L  =  0,016,  and  the  weight  of 

the  cubic  foot  of  water  66  Ibs.,  for  which  the  student  can  at  pleasure  substitute  0,0155 
and  62,5  respectively,  also  8,02  for  7,906.— AM.  ED.] 


276     •  TURBINES  WITHOUT  GUIDE-CURVES. 

angle  fl  =  150°  to  160°,  in  order  to  have  x  as  low  as  possible.  The 
ratio  i  =  —  is  only  1,15  to  1,30,  as,  otherwise,  from  fl  being  so 

large,  the  length  of  bucket  would  be  inconvenient.  In  order  to  have 
the  loss  of  mechanical  effect  for  the  entrance  on  the  wheel  as  low  as 
possible,  the  water  is  laid  on  to  the  wheel  with  a  velocity  of  only 
2  feet,  and  hence  the  internal  radius  r1  is  only  Q,4  \/Q,  and  the 
external  radius  r  =  v  rl  =  0,4  v  </Q. 

The  best  velocity  of  rotation  is  also  to  be  calculated  by  a  different 
rule,  as  the  formula : 

/  2#A 

Vl ~ \/  a.m.^r.      /_^n.j_v>      AT y 

V        sin.  (3  — a)   ^   *  \sin.  (fl  —  «)/  \rj 

which  in  this  case  has  the  form: 


.  tang.  /32  +  *  (L\* 


gives  too  great  values.  The  reason  of  this  is,  that  the  condition  of 
making  the  velocity  of  discharge  =  0,  does  not,  on  account  of  the 
prejudicial  resistances,  lead  to  the  maximum  effect  being  produced  ; 
and  it  is  only  for  turbines  with  guide-curves,  that  the  fulfilment  of 
this  condition  gives  satisfactory  approximations  to  this  maximum. 
On  the  other  hand,  for  turbines  without  guide-curves,  and  in  all 
cases  in  which  o  is  nearly  90°,  the  influence  of  the  prejudicial  re- 
sistances on  the  working  of  the  wheel  becomes  too  great  for  its 
being  possible  to  assume  that  w  =  0,  or  v  =  c2.  In  order  to  find 
the  least  velocity  for  these  wheels,  we  adopt  the  following  method  : 
We  have  already  (Vol.  II.  §  147)  found  that 

(1  +  ,)  c22  =  2gh  +  v2  —  2  c  vl  cos.  o—  f  c2,  and  as 

cos.  o  =  cos.  90°  =  0,  and 
sin.  r  put: 


sin.  (,3  —  90) 

(1  +  x)  c22  =  2gh  +  v2  fl  —  S(fy* 
and,  therefore,  the  velocity  of  discharge  : 


i      /  2gh  +  ^  fl  —  S 


*  tang. 


1  + 
Hence,  the  loss  of  fall  : 

c22  +  v*  —  2  c2  v  cos.  8  -f  x 


(1  +  «)  c22  +  v2  fl  +  f  p-V  tang.  ff\  —  2  v  c2  cos.  8 

_ 


TURBINES  WITHOUT  GUIDE-CURVES.  277 


and,  therefore,  the  effect  to  be  expected  from  the  wheel  : 


L=    v  cos.  8 

V 


If  we  put  4-  for  1  —  ?  (!l\2  tawgr,  02,  and  t  for  ^    +  *,  then  we  have, 
\  r  /  cos.  6 

more  simply,  L  =  (v  </2gh  +  *  v2  —  $  v2)  ~^-. 

*# 
In  order  that  this  value  may  give  a  maximum,  we  can  deduce  by 

the  higher  calculus  that  <j»  v  =     s    +jM|  —  ^  or  jf  we  represent  the 


ratio  of  the  height  due  to  velocity  ?L  to  the  pressure  height  A,  that 
is,  A  by  *,  then  *  +  **  =  t|  and  hence,  ^  =  »-^^E*.     If 


from  this  we  have  got  *,  we  have  for  the  velocity  v  =  -Sx  -  2gh, 

r.  ,  \2sh  +  4-  v2     TT         j.1. 

vl  =  —  v.  c  =  — v.  tana.  p,  and  <?„  =    I  -5 — '     — .    Hence  the  sec- 
r  \     1  +  * 

tions  jP=  — ,  and  F3  =  _r,  and,  lastly,  the  height  of  the  wheel,  or 

of  the  orifice  e  = . 

The  other  proportions,  as  the  construction  of  the  buckets,  &c.  &c., 
do  not  differ  from  those  of  turbines  having  guide-curves. 

Remark.  Strictly  speaking,  turbines  with  guide-curves  should  also  be  treated  in  this 
manner,  but  as  the  expressions  are  very  complicated,  and  lead  to  a  value  of  — ,  which 
differs  very  little  from  unity,  we  have  deemed  the  investigation  unnecessary. 

Example.  It  is  required  to  make  the  necessary  calculations  for  the  design  of  a  turbine 
on  Cadiat's  plan,  for  a  fall  of  5  feet,  with  SO^ubic  feet  of  water  per  second.  Assuming 
B=  150°,  »=  1,2,  and  r,  =  0,4  ^  =  0,4  ^/3Q  =  2,19,  which  we  shall  make  2,25,  and, 
hence,  r=  1,2  .  2,25  =  2,70  feet.  If,  further,  f  =  0,1 5,  and  *  =  0,10,  and  >=  16°,  then 
.  =  t  _£  /!jV  .  tang.  &=  1  —  0,15  .  (<a"S-  30°)'  =  i  _ 0,035  =  0,965, and 


_       __  vi      _  1091  ;  and,  therefore, 
"    co*.  J-         co».16° 


_  +  -^-4«        1,091-0.475  and  ^-_  0)820. 

24v/t7::::T1          1,93.0,475 

VOL.  ii.—  24 


278  WHITELAW'S  TURBINES. 

From  this  we  have  the  most  advantageous  velocity  of  rotation  : 

1 


v  =  ^/x^2gh  =  0,82  .  7,906  v/5~=  14,50  feet.     Again,  vt  =  -=  14'5°  =  12,08  feet, 


c  =  —  v,  tang.  8  =  12,08  tang.  30°  =  6,97  feet,  and 

pr+T^=     /318.5  +  202.9  _21>65feet 

•  w  i+«      w     1,1 

and  now  we  have  the  section  F  =  —  =  -  =  4,304   square   feet,  and    the    section 
c        6,97 

F  2  =  —  =  -  =  1  ,386  square  feet.     Hence,  again,  we  have  the  height  of  the  wheel  : 

t  —  -  =  —  '  -  =  0,304  feet,  and  if  we  take  for  the  orifices  of  discharge  of  the 
2*r,       2.  2,25.  » 

wheel,  the  proportional  dimensions  —  =  f  ,  we  have  for  the  number  of  buckets: 

n  =  L?A  —  2  '  1;386  as  2'772    =  30.     If  the  thickness  of  thebucket  plates  s  =  0,017 

e*  0,304'  0,0924 

feet,  we  have  as  the  angle  of  discharge  : 

•     >       F2—ne*       1,386—30.0,304.0,017  1,226 

sin.  c  —  gg-  —  -  -  —  --    gg-7  -  -  —  U.-ioo, 
2  w  r  e  2  .  2,7  .  0,304  v  1,6416  «• 

and,  hence,  »  =  1  3|°.     As  we  assumed  above  that  for  4  =v/t  +_  ,  >=  16°,  the  velo- 

cos.  > 

cities,  sectional  areas,  &c.,  just  found,  will  be  slightly  varied  by  the  introduction  of 
>=  13f  °.     The  efficiency  of  this  wheel  is  : 


—  (\  —  (v_  ,n—  " 

^ 


156'25          ;  1,116.  3,049 


_  1  —  0,321        0,679  _  tho  coefficient  7  Doe  being  taken  for  Prussian  measures, 

1,17  1,17 

as  in  last  example. 

For  the  same  fall,  a  Fourneyron's  turbine  gave  an  efficiency  «  =  0,705.  (See  example 
to  last  paragraph.) 

§  160.  Whitelaw's  Turbines.  —  The  Scottish  turbine  has  to  be 
treated  differently  from  that  of  Cadiat,  inasmuch  as  the  water 
enters  the  wheel,  in  great  measure,  in  a  manner  involving  shock, 
and  because  in  these  turbines  the  dimensions  and  form  of  the  wheel- 
channels  are  much  more  arbitrary  than  for  the  other.  The  angle  8 
may  be  made  much  less  in  these  than  in  the  other  forms  of  turbine. 
They  are  peculiarly  adapted  for  falls  of  great  height,  with  small 
supply  of  water. 

The  width  of  the  pressure  pipe  may  be  determined  by  the  condition 
that  there  shall  be  a  maximum  velocity  of  6  feet  per  second  through 
it.  So  that  the  internal  radius  of  the  wheel,  or  the  radius  of  the 

pressure  pipe  rl  =     ITT^—  =  0>23  \/  Q.    The  external  radius  is  made 

2,  3,  or  4  times  this,  according  as  the  number  of  arms  or  discharge 
channels  is  4,  3,  or  2.  The  velocities  v,  vv  and  c,  and,  therefore/  the 
sections  jP:  and  -F2,  may  be  determined  as  in  the  case  of  turbines 
without  guide-curves  (last  paragraph).  The  depth  or  height  of  the 

F  F 

wheel  e  =  -  ,  and  the  width  of  the  orifices  of  discharge  =  d  =  —  £. 
2  n  rl  ne 

v2 
In  determining  v  or  x  =  -  »  it  will  be  necessary  to  take  ?  higher 

than  0,15,  as  shock  cannot  be  avoided  where  the  stream  of  water 


WHITELAW  S  TURBINES. 


279 


divides  itself  to  run  in  so  many  different  directions.  It  may  be 
assumed,  without  much  risk  of  error,  that  £=  0,20.  As  the  arms  or 
wheel  channels  are  considerably  longer  than  in  any  other  turbine, 
we  must  take  a  higher  value  for  *,  and  assume  it  at  least  0,15. 

The  arms  are  generally  curved  according  to  the  Archimedian 
spiral  ;  they  may  be  curved  to  the  form  ABD,  Fig.  272,  composed 
of  two  circular  arcs,  AB  and  BD.  For  this,  the  orifice  of  the  inlet 
pipe,  or  internal  periphery  of 

the  wheel,  is  divided  into  as  Fig.  273. 

many  equal  parts  as  there  are 
to  be  arms  —  three  in  the  case, 
Fig.  273,  and  from  each  of 
these  draw  the  line  AS,  mak- 
ing the  angle  /3  with  the  tan- 
gent at  that  point  ;  or,  for 
example,  make  SAO 
=  270°  —  0°  =  90°  +  13°,  then 
with  the  external  radius  r,  de- 
scribe a  circle,  and  divide  it 
into  as  many  equal  parts  as 
there  are  to  be  arms,  but  so 
that  between  the  two  points  A 
and  D  of  the  two  peripheries, 
a  central  angle  of  about  135°, 
150°,  or  180°  is  included,  according  as  the  number  of  arms  is  4,  3, 
or  2.  The  direction  of  the  axis  DT  being  laid  off  in  such  manner 
that  the  angle  CD  I7  equals  about  80°,  we  find  the  centres  M  and  0 
of  the  arcs  AB  and  BD  forming  the  axis,  by  bisecting  the  angles 
SAD,  TDA  by  the  straight  lines  AB  and  BD  ;  then  draw  ST 
parallel  to  AD,  and  AM  at  right  angles  to  AS,  BO  at  right  angles 
to  ST,  and  DO  at  right  angles  to  DT.  We  see  the  reason  of  this 
at  once,  if  we  consider  that,  by  division  of  the  angles  SAD  and 
TDA,  and  by  drawing  the  parallel  ST,  the  angles  MBA  and  MAB, 
and  also  the  straight  lines  MA  and  MB,  are  made  equal  to  each 
other,  that  in  like  manner  the  angles  ODB  and  OBD,  as  also  the 
lines  OB  and  OD,  are  made  equal  to  each  other. 

To  find  the  outsides  of  the  pipes,  DGr  is  made  =  Dfl"=  the  half 

width  of  orifice  -,  and  FN  is  made  =  KN,  and  the  arcs  HK  and 
GrF  are  drawn,  so  that  the  width  GrH  gradually  passes  into  FK,  &c. 

Example.  Required  to  design  a  Scottish  turbine  for  a  fall  of  150  feet,  with  a  supply  of 
water  of  l£  cubic  feet  per  second.  In  the  first  place,  the  internal  radius 
r}  =  0,23  v/Q  =  0,23  ^/  l^5~  =  0,282  feet  ;  but  we  shall  put  it  =  0,3  feet,  and  the  diameter 
of  the  pressure  pipe  9  inches,  or  0,75  feet;  we  shall  have  only  two  arms,  and  make  the 
external  radius  r  =  4  .  rt  =  1,2  feet.  We  shall  put  B  =  150°,  and  »  =  10°,  and  assume 
«  ==  0,15,  and  £  =  0,25,  hence  : 

1  —0,25  .  ^  (tang.  30°)'=  1  —  0,0052  =  0,994  8,and 


—0,25  .  ir-lV 


tang. 


cos.  10° 


1,0890. 


280  COMPARISON  OF  TURBINES. 

Of  the  fall  h  =  150,  the  friction  of  the  water  in  the  9  inch  pipe,  which  may  be  presumed 
to  be  200  feet  long,  consumes,  according  to  Vol.  I.  §§  331  and  332,  an  amount : 

z  =  0,0213  .  0,016  .  (-V  .  ISi.  =  0,0003408  .    flY  .  2°°  '  K5' 
\w/        d?  \w  /  0,75s 

=;  0,0003408  .  1,621     20°  •  256  _  n  n.-una  .  i  621  .  *'*  =  1.05  feet;  therefore,  we 

27 

must  only  introduce  h=  148,95  feet  in  our  calculations.     For  the  most  advantageous 
velocity  : 

_<?  —  VV  —  4-  _  1,089  —  ^/1,1858  —  0,9948  _  1,089  —  0,437 
*~2~7^/*^?~  1,9896  v'oTm 0,8695 

And  hence : 

r  =  ^  .  2gk  =  ^0,75  .  62,5  .  148,95  =  83,56  feet,  t>,  =  -  =  20,89  feet; 
c  =  —  r,  tang.  0  =  20,89  tang.  30°  =  12,06  feet ;  and 

694 5'8  =  118,89  feet.     (Prussian.) 
1,15 

From  this  we  have  the  sections:  F  =  _  = ! =  0,1244  square  feet,  and 

c         12,06 

.Fa  =  _  =— — =0,01262   square  feet     From  this  again  we  have  the  height  of 
ca       118;82 

wheel  e  =  -?-  —  °'1244=  0,066,  and,  lastly,  the  width  of  orifice: 
2wr.  —    0,6  * 


d=  gg.=  0.01262  __^"  •*"«_.  QQ956    feet  =  1,15  inches.     In  order    to   have  a 
ne        2  .  0,066          0,132 

greater  ratio  —  between  the  sides  of  the  orifices,  we  should  have  to  introduce  more  arms 

a 

or  wheel  channels ;  but  as  the  channels  are  very  long,  even  the  above  proportion  would 
be  found  to  insure  &  full  flow  through  them.  The  efficiency  of  the  wheel,  neglecting  the 
friction  at  the  joint  and  losses  in  the  pressure  pipe,  is  : 

1 


=  (1  —  0,191  .  1,5) I __0,7135__       n 

'  1,089  .  1,07        1,167 

§  161.  Comparison  of  Turbines. — Let  us  now  draw  a  comparison 
between  the  three  turbines  of  Fourneyron,  Cadiat,  and  Whitelaw. 
The  turbine  with  guide-curves  is  unquestionably  the  more  perfect 
construction,  mechanically  considered,  as  by  this  arrangement  (when 
c2  =  r)  the  entire  vis  viva  of  the  water  may  be  taken  from  it,  which 
cannot  be  done  without  this  apparatus.  All  things  considered,  the 
velocity  of  rotation  for  all  the  wheels  is  nearly  the  same,  viz.  : 
r3  v  =  0,7  *S2gh  to  \/2gh  for  the  maximum  effect.  This  maximum 
effect  is  nearly  the  same  for  each  of  them,  the  advantage  being  on 
the  side  of  Fourneyron's  wheels,  when  working  in  its  normal  state, 
and  on  the  side  of  Whitelaw 's,  when  the  supply  of  water  is  very 
variable.  The  Scottish  turbine  may  be  constructed  at  less  cost  than 
Fourneyron's  turbines  with  guide-curves. 

In  general  terms,  we  believe  that  the  turbines  of  Fourneyron  and 
Cadiat  are  better  adapted  for  very  low  falls  and  those  of  moderate 
height  (up  to  30  feet)  with  large  supplies  of  water,  whilst  for  high 
falls  and  small  supplies  of  water,  Whitelaw's  wheels  are  to  be 
preferred. 


EXPERIMENTS  ON  TURBINES.  281 . 

Remark.  In  the  case  of  turbines  without  guide-curves,  especially  when  the  fall  is  high, 
the  water  leaving  the  wheel  retains  a  consider- 
able absolute  velocity  w  =  ca  —  »,  and  hence  a  Fig.  274. 
notable  amount  of  vis  viva  is  lost.  This  loss  may 
be  avoided,  or  at  least  much  diminished,  if  the 
vis  viva  of  the  water  leaving  the  turbine  be  applied 
to  a  second  wheel.  M.  Althans,  of  the  Sain  Iron 
Works,  has  put  this  into  practice  at  a  mill  near 
Ehrenbreitstein.  The  essential  part  of  the  con- 
struction of  this  wheel  is  represented  by  Fig.  274. 
JlEJl  is  a  reaction  wheel  with  four  curved  dis- 
charge pipes,  the  fall  being  120  feet  (compare 
§  147),  BB  is  a  larger  wheel  with  curved  buckets, 
set  in  motion  by  the  water  discharged  at  A,  A. 
As  the  wheels  revolve  in  opposite  directions,  they 
have  to  be  connected  with  each  other  by  revers- 
ing gearing.  The  outer  wheel  has  this  further 
advantage,  that  it  adds  to  the  fly,  or  regulating 
power  of  the  machine.*  (See  l<  Inner- 6s  terrei- 
chisches  Gewerbeblatt,"  Jabrgang  5,  1843.) 

§  162.  Experiments  on  Turbines. — Numberless  experiments  on . 
turbines  of  the  different  forms  we  have  now  been  discussing  are 
extant,  but  the  reported  results  are  not  all  trustworthy.  These 
recipients  of  water  power  are  in  many  respects  admirable  machines, 
but  to  suppose  that  an  efficiency  =  0,85  to  0,90  has  been  obtained 
from  them,  arises  from  some  mistake.  As  the  discharge  of  water 
through  the  most  perfectly  formed  orifice  has  a  velocity  co-efficient 
V  $  =  0,97  (Vol.  I.  §  312),  there  must  be  a  loss  of  mechanical  effect 
at  entering  the  wheel,  represented  by 

1          \  c2  c2 
1)  —  Q  y  =  0,06  —  Qy.     As,  again,  the  friction  of  water 

d>2  /  2p"  2fi° 

in  a  pipe  six  times  as  long  as  it  is  wide,  consumes  (Vol.  I.  §  331) 
0,019  .6  .  —  Qy  =  0,114  —  Qy,  or  11,4  per  cent,  of  the  avail- 
able fall  (as  —  =s  —  nearly  =  h\  we  see  that,  deducting  these  re- 
sistances, there  remain  only  83  per  cent,  of  effect  over.  If  we 
allow  only  2  per  cent,  for  the  resistance  in  the  curved  conduits,  2 
per  cent,  for  shock  on  the  ends  of  the  buckets,  and  3  per  cent,  for 
the  mechanical  effect  retained  by  the  water  discharged,  and  neglect- 
ing all  other  sources  of  loss,  such  as  is  involved  in  the  guide-curves, 
&c.,  there  remain  only  76  per  cent,  of  useful  effect,  and,  therefore, 
a  turbine  that  gives  us  an  efficiency  ij  =  0,75,  may  be  considered  as  a 
very  excellent  one.  The  experiments  of  Morin  and  other  impartial 
persons  give  results  as  to  efficiency  as  high  as  0,75,  but  never  above 
this. 

Morin's  experiments  were  published  about  ten  years  ago,  under 
the  title  "  Experiences  sur  les  roues  hydrauliques  a  axe  vertical, 

*  [A  second  wheel  to  receive  the  water  frpm  a  common  Barker's  mill  was  used  in 
model  by  the  Editor,  to  illustrate  his  lectures  before  the  Franklin  Institute,  about  the  year 
1830-31.  The  model  is  probably  now  at  Carlisle,  Pa.— AM.  Eu.J 

24* 


282  EXPERIMENTS  ON  TURBINES. 

appcle'es  Turbines,  Metz  et  Paris,  1838."  The  first  experiments 
were  made  on  one  of  Fourneyron's  turbines  at  Moussay,  external 
diameter  of  wheel  =  2,8  feet,  depth  =  0,36  feet,  fall  =  24,6  feet, 
and  the  quantity  of  water  laid  on  =  26  cubic  feet  per  second. 
Thus  there  was  a  fall  of  upwards  of  70  horse  power  at  disposition. 
The  result  of  these  experiments,  stated  in  general  terms,  was  that, 
whether  the  wheel  worked  more  or  less  in  back-water,  it  gave  for 
180  to  190  revolutions  per  minute,  a  maximum  effect  of  69  per  cent. 
of  the  whole  power.  When  the  number  of  revolutions  was  greater 
or  less  by  from  40  to  50  per  cent,  of  the  above,  the  efficiency  was 
from  7  to  8  per  cent.  less.  These  were  the  results  when  the  cylin- 
drical sluice  was  quite  drawn  up  ;  but  when  the  sluice  was  lowered 
to  half  the  height  of  the  wheel,  the  efficiency  was  reduced  about  8 
per  cent.  Had  the  wheel  been  entirely  free  of  back-water,  this  fall- 
ing off  in  efficiency  must  have  been  greater. 

Experiments  on  a  turbine  at  Muhlbach  for  a  fall  of  120  horse 
power,  gave  the  following  results  :  diameter  of  wheel  2  metres, 
height  ^  metre,  fall  12  feet,  with  86  cubic  feet  of  water  per  second. 
With  the  sluice  quite  drawn,  the  wheel  made  50  to  60  revolutions, 
and  the  efficiency  was  0,78  according  to  Morin  ;  but  he  has  adopted 
too  low  a  co-efficient  of  discharge  in  calculating  the  quantity  of 
water,  and,  therefore,  0,75  is  probably  the  true  efficiency.  For 
variations  of  from  30  to  80  revolutions,  the  efficiency  did  not  vary 
more  than  4  per  cent,  from  the  above.  The  efficiency  was  the  same 
whether  the  wheel  was  only  a  few  inches,  or  3  feet  under  water. 
The  efficiency  was  nearly  constant  for  great  variations  in  the  quan- 
tity of  water  laid  on.  As  the  sluice  was  depressed,  the  efficiency 
fell  off  rapidly.  Morin  directed  experiments  to  ascertaining  the 

ratio  —  -  —  ,  and  found,  as  theory  indicates,  that  this  ratio  increases 


as  v  increases  (owing  to  the  influence  of  centrifugal  force),  and  de- 
creases as  the  sluice  is  raised. 

§  163.  Redtenbacher  gives  the  result  of  some  experiments  on 
turbines  in  Switzerland,  in  his  work  "  Ueber  die  Theorie  und  den 
Bau  der  Turbinen  und  Ventilatoren."  These  were  ill  constructed, 
and  gave  low  results. 

Among  other  interesting  results  which  Redtenbacher  deduces 
from  the  recorded  experiments  on  Fourneyron's  turbines,  we  may 
particularly  mention  that  these  wheels,  when  working  with  their 
maximum  effect,  and  with  sluice  fully  drawn,  make  half  the  number 
of  revolutions  that  they  do  when  working  free  of  all  load  but  their 
own  inherent  resistances. 

Combes'  experiments,  with  models  of  his  wheels,  give  less  efficiency 
than  those  above  mentioned,  viz  :  0,51  to  0,56. 

Mr.  Ellwood  Morris,  of  Philadelphia,  has  recorded  a  very  complete 
set  of  experiments  on  two  turbines  of  Fourneyron,  (see  "  Journal  of 
the  Franklin  Institute,"  Dec.  1843.) 

One  wheel  was  4f  feet  diameter,  8  inches  high,  6  feet  fall,  1700 


FONTAINE'S  TURBINE.  283 

cubic  feet  of  water  per  minute.  Sluice  drawn  6  inches,  52  revolu- 
tions per  minute,  efficiency  found  to  be  0,7.  The  velocity  vl  of  the 
inner  periphery  of  the  wheel_was  then  =  0,46  ^/2gh.  For  varia- 
tions between  vl  =  0,5  ^/~2gh  to  0,9  ^/2g~h,  the  value  of  q  varied 
from  0,64  to  0,70.  The  other  wheel  was  4'  —  5"  diameter,  6  inches 
deep,  4J  feet  fall,  14  cubic  feet  per  second.  It  revolved  under 
water,  and  when  the  sluice  was  drawn  4J  inches,  the  effects  were 
as  follows  :  For  vl  =  25  to  30  per  cent,  of  </  2gh,  then  n  =  0,71. 
0,45,  that  is  u  =  49,  the  maximum  effect  was  obtained, 


or  n  =  0,75.     For  —  ^—  =  0,5  to  0,7,  the  value  of  «  =  0,60. 


Remark.  The  results  of  experiments  on  Cadiat's  wheels  are  probably  overstated.  Ex- 
periments on  Whitelaw's  wheels,  as  made  by  Messrs.  Randolph  and  Co.,  of  Glasgow, 
give  results  varying  from  0,60  to  0,75  for  the  efficiency. 

§  164.  Fontaines  Turbine.  —  The  turbines  recently  introduced  by 
Fontaine  and  Jonval,  differ  from  those  of  Fourneyron,  inasmuch  as 
the  guide-curves,  instead  of  being  in  one  place  with,  are  placed  above 
the  wheel,  and  thus  the  water  does  not  flow  outwards  on  to  the 
wheel,  but  from  above  downwards,  and  is  discharged  from  the  bot- 
tom of  the  wheel.  Centrifugal  force  plays  only  a  very  subordinate 
part  in  the  motion  of  the  water  through  these  wheels,  gravity  taking 
its  place.  The  difference  between  the  turbine  of  Fontaine  and  that 
of  Jonval  consists  in  the  former  being  placed  immediately  on  or 
under  the  surface  of  the  water  in  the  race;  while,  in  Jonval's  arrange- 
ment, the  water  flowing  through  the  wheel  forms  a  column  of  water 
under  the  wheel,  but  acts  upon  the  wheel  just  as  if  it  pressed 
upon  it.  The  arrangement  of  Fontaine's  turbine  is  shown  in  Fig. 
275,  in  vertical  section  and  in  plan.  AA  is  the  wheel,  BB  is  the 
wheel  plate,  uniting  the  wheel  with  the  hollow  axle  OODD.  In 
order  that  the  pivot  may  be  out  of  the  water,  the  axle  CD  ends  in 
an  eye  GrGr,  in  which  there  is  a  steel  plug  FS  (which  can  be  raised 
or  depressed  by  the  screw  $)  resting  on  the  solid  axle  EF  at  F. 

The  motion  of  the  wheel  is  transmitted  by  an  axle  H,  firmly  con- 
nected with  the  head  of  the  hollow  shaft.  *To  keep  the  upright 
shaft  from  the  water,  it  is  surrounded  by  a  casing,  as  in  Fourney- 
ron's  turbines.  The  guide-curve  apparatus  KKis  screwed  on  to  the 
beams  LL,  and  to  it  there  is  a  plate  KMMK  united,  having  a  cylin- 
drical metallic  bed  M  M,  in  which  there  is  a  'collar  similar  to  that 
at  DD,  for  maintaining  the  perpendicularity  and  steadiness  of  the 
shaft.  The  form  of  the  guide-curves  JV,  and  of  the  wheel-buckets 
0,  is  represented  at  III.  For  regulating  the  quantity  of  water  laid 
on,  there  is  a  compound  sluice,  having  as  many  separate  valves  as 
there  are  guide-curves.  These  valves  are  covered  by  round  pieces 
of  wood,  and  are  fastened  by  screws  and  nuts  to  the  cylindrical 
casing  of  the  guide-curve  apparatus.  The  sluice-rods  PQ,  PQ  .  .  . 
are  firmly  united  to  each  other  by  an  iron  ring  QQ,  which  can  be 


284 


FONTAINE'S  TURBINE. 

Fig.  275. 


raised  and  depressed  by  three  lifting-rods  QR,  QR  .  .  .  For  this 
purpose  the  ends  .R,  R  .  .  .  of  these  rods  are  cut  as  screws,  and 
toothed  wheels  T7,  T .  .  .  put  on  them,  the  nave  or  box  of  these 
having  female  screws,  and  the  peripheries  of  the  whole  being  en- 
compassed by  an  endless  chain. 

When  one  of  these  wheels  is  moved  by  means  of  a  winch,  or  other- 
wise, it  is  evident  that  the  rest  must  be  so  too,  so  that  the  three  rods 
are  moved  simultaneously. 


JONVAL'S  TURBINE. 


285 


§  165.  Jonval's  Turbine.— Figs.  276  and  277,  I,  II,  and  III,  re- 
present Jonval's  turbine.     Here,  again,  J1A  is  the  wheel,  united  to 


Fig.  276. 


the  upright  shaft  CD  by  a  disc  or  plate ;  BB  is  the  guide-curve 
apparatus  opening  as  a  diverging  cone  into  the  lead.  The  pivot 
rests  on  a  footstep  C1,  supported  by  EE.  The  relative  position  of 
the  wheel  and  guide-curves,  as  also  a  part  of  the  outside  of  the  pipe 
in  which  the  wheel  is  enclosed,  is  represented  at  II  and  III. 

To  keep  the  surface  of  the  water  in  the  lead  free  from  agitation, 
a  float  SS  is  placed  on  it,  and  for  regulating  the  wheel's  motion,  a 
sluice  G  is  introduced,  worked  by  a  handle  at  K.  According  as 
this  sluice  is  raised  or  depressed,  more  or  less  water  flows  away,  and 
thus  the  power  is  regulated. 

The  framing  LL  supports  the  plumber-blocks  for  the  upper  end 
of  the  shaft,  and  for  a  horizontal  shaft,  through  which  the  motion 
is  first  transmitted  by  a  pair  of  mitre  wheels.  When  the  wheels  are 
small,  the  reservoir  or  well  in  which  the  wheel  is  enclosed  may  be 
of  cast  iron ;  for  large  wheels  it  should  be  built  of  solid  masonry. 

It  is  evident,  from  what  we  have  now  detailed,  that  the  turbines  of 
Fontaine  and  Jonral  are  essentially  alike  in  their  main  proportions, 
and  that  their  theory  is  the  same.  In  both,  the  water  in  the  lead 
stands  at  a  certain  height  h1  above  the  point  of  entrance  on  the 
wheel.  The  water  in  the  race,  however,  stands  in  Jonval's  turbine 


286 


FONTAINE'S  AND  JONVAL'S  TURBINES. 


at  a  certain  depth  h2  below  the  wheel,  while,  in  Fontaine's  arrange- 
ment, the  race-water  is  in  immediate  contact  with  the  wheel.  The 
regulation  of  the  wheel's  motion  is  managed  in  Fontaine's  by  an 

Fig.  277. 


internal,  and  in  Jonval's  by  an   external  sluice  —  the  one   being 

analogous  in  this  respect  to  Four- 
Fig-  278-  neyron's  turbine,  the  other  to  that 
of  Cadiat. 

§  166.  Theory  of  Fontaine's  and 
Jonval's  Turbines. — In  developing 
the  theory  of  the  turbines  of  Fon- 
taine and  Jonval,  we  shall  adopt 
the  following  symbols. 

Let  the  angle  of  inclination  of  a 
guide-curve  NG  to  the  horizon,  or 
the  angle  of  entrance  of  the  water 
JYGG,  =  c  Av,  Fig  278,  =  a.  The 
angle  cl  Jlv  which  the  upper  end  of  the  wheel  bucket  Jl  makes  with 
the  motion  of  the  wheel  =  /3,  and  the  angle  DD^,  at  which  the 
bottom  of  the  wheel  bucket  meets  the  horizontal  =  8.  Let  the  abso- 
lute velocity  of  entrance  of  the  water  on  the  wheel  Jlc  =  c,  the  mean 

radius  of  the  wheel  r  =  TI  ^    r* ,  corresponding  to  the  velocity  of  the 
wheel  Av  =  v.  The  relative  velocity  of  entrance  Acl  =  cv  and  the  velo- 


FONTAINE'S  AND  JONVAL'S  TURBINES.  287 

city  of  discharge  Bc2  =  c2.  Again,  let  F=  the  sum  of  the  areas  of  all 
the  sections  JVGj  of  the  water  flowing  out  of  the  guide-curve  apparatus, 
Fl  the  sum  of  the  upper  sections  GjjfiT,  and  F2  the  sum  of  the  lower 
sections  DE  of  the  wheel  channels. 

If,  again,  ?  be  the  co-eificient  of  resistance  in  the  guide-curve 
canals,  and  x  the  head  measuring  the  pressure  of  the  water  entering 
the  wheel,  then  (1  +  £)  c2  =  2g  (A:  —  x) ;  and  reckoning  the  height 
a  (34  feet)  of  a  column  of  water  equal  to  the  atmospheric  pressure, 
then  (1  +  £)  c2  =  2g  (a  +  h,  —  x). 

For  the  relative  velocity,  we  have: 

c*=  c2  +  v*—2cvcos.  o. 

If,  again,  b  —  the  depth  of  the  wheel,  y  =  the  height  of  a  column 
of  water  =  to  the  pressure  of  water  immediately  under  the  wheel, 
and  x  the  co-efficient  of  resistance  in  the  wheel  channels,  then,  for 
the  relative  velocity  of  discharge,  we  have: 

(1  +  *)c22  =  2g  (b  -f  x  —  y)  +  c*  =  2g(a+h1  +  b  —  y) 
+  v*  —  2cv  cos.  a  —  £  c2 

If  we  here  again  endeavor  to  take  from  the  water  as  mrfch  effect 
as  is  inherent  in  it,  and,  therefore,  make  c2  =  v,  and  also 

c  =      v  sm"  **    ,  we  then  have  for  the  relative  velocity  of  discharge: 
sin.  (ft  —  a) 

rin.fi  ca...         /j giPy  +  ."I*.  2g(a  +  hl  +  6-y), 

Sin.  (ft  —  a)  \nn.  (0  —  a)/  J 

and,  therefore,  the  best  velocity  of  the  wheel: 
2g  (a  +  At  +  b  —  y) 


'^ +  £(*"•  (*-"))'  +  . 
-o)  \(Sin.  ft —  a)/ 


The  pressure-height  y,  when  the  turbine  revolves  in  free  air,  is 
equal  to  the  atmospheric  pressure  a;  but  when  the  turbine  is  in  back 
water,  it  =  a  -f  A2,  where  h2  is  the  height  of  the  surface  of  the 
water  above  the  bottom  of  the  wheel ;  and  lastly,  when  the  wheel  is 
above  the  race  water,  as  in  Jonval's  arrangement,  y  =  a  —  A2  +  -, 
where  h2  =  the  depth  of  the  race  surface  underneath  the  bottom  of 
the  wheel,  and  z  is  the  height  due  to  the  velocity  of  the  water  flow- 
ing through  the  sluice  from  the  reservoir  to  the  tail-race.  The  total 
fall  for  the  case  of  the  wheel  revolving  free  of  back  water  is 
A  =  Aj  -f  b;  when  the  wheel  is  in  back  water,  h  =  A,  +  b  —  h2; 
and  when  the  wheel  is  above  the  tail  water,  h  =  ^  -f  b  +  hr  Hence, 
for  the  two  first  cases : 


%A 

— i 


«n.  0  COS.  a    ,     y  /        SWl.  o        \      , 

o •  -|-  i  I 1 

sin.  (J3  —  a)         \«n.  (j3  —  o)/ 


and,  for  the  latter: 


sin.  P  cos,  a         /     *»» 


288  FONTAINE'S  AND  JONVAL'S  TURBINES. 

and,  when  the  orifice  Gr  by  which  the  vessel  communicates  with  the 
tail-race  is  large,  or  when  the  water  flows  away  very  slowly  : 


§  167.  From  the  velocity  v  =  c2,  the  absolute  velocity  of  entrance 

v  sin.  8  ,  ..  !•  i  ,. 

c  =  --  —  ,  and  the  pressure  height  : 
sin.  (/3  —  a) 


may  be  calculated.     Neglecting  prejudicial  resistances  : 

,  h  sin.  0 

2  cos.  o  sin.  (/3  —  a)' 
and  neglecting  the  atmospheric  pressure  : 


2  cos.  o  sin.  ()3  —  o) 
x  =  0,  05  more  correctly  x  =  the  external  pressure  of  the  atmosphere 

when  h.  = '— .     The  loss  of  water  involved  in  the 

2  cos.  a  sin.  (j3  —  o) 

free  space  necessarily  left,  depends  on  the  difference  between  the 
internal  pressure  (a;),  and  the  external  pressure  at  this  point,  and  is 
different  in  the  two  turbines  now  under  consideration.  That  the 
water  may  flow  on  in  a  connected  stream,  x  must  never  descend  to  0, 

that  is,  we  must  have  a  -f  h.  > .   Again,  that  the 

2  cos.  o  sin.  (/3  —  o) 

water  may  not  recede  from  the  bottom  of  the  wheel,  we  must  never 
have  y  =  0,  that  is,  we  must  have : 

a  —  hz  +  z  =>  0,  or  A2  <  a  +  z,  or  A2  <:  a  +  —  ( 

Hence,  when  the  area  of  the  orifice  G-  is  large,  we  must  have 
A,  <  a.  From  this  we  see  that  the  height  of  the  wheel  above  the 
surface  of  the  tail-race  must  never  reach  to  the  water-barometric 
height  of  34  feet. 

If,  in  Jonval's  turbine,  the  reservoir  be  high  and  narrow,  so  that 
the  velocity  of  the  water  in  it  is  considerable,  there  arise  losses  of 
effect  at  this  point  from  friction,  resistance  in  curves,  impact,  &c., 
&c.  On  this  account,  it  is  advisable  to  make  the  reservoir  wide  in 
proportion  to  the  wheel's  diameter. 

§  168.  The  Mechanical  Effect  of  Fontaines  and  JonvaTs  Tur- 
bines.— The  effect  of  these  turbines  may  be  deduced  exactly  as  that 
of  Fourneyron's  has  been,  by  subtracting  from  the  total  mecha- 
nical effect  ofrthe  fall  Q  hy,  the  effect  consumed  by  different  pre- 
judicial resistances,  &c.  The  loss  in  the  guide-curve  apparatus 

_2  p2 

Ll  =  £  .  —  Qy,  and  that  in  the  wheel-channel  L2  =  *  —  Q  y.    Again, 

the  loss  of  the  vis  viva  retained  by  the  water  at  its  exit  from  the 
wheel,  is 


FONTAINE'S  AND  JONVAL'S  TURBINES.  289  • 


In  Jonval's  turbines,  there  has  to  be  added  to  these  the  loss  of 
effect  involved  in  the  velocity  of  discharge  (w.)  through  the  sluice 


_. 

2g         ~2g     G* 
Hence  the  total  effect  of  the  wheel  : 


We  see  from  this  that  the  loss  of  effect  increases  as  the  angle  8  in- 
creases, and  as  the  velocity  wl  is  greater,  or  as  the  velocity  of  dis- 
charge and  sluice-opening  Gr  are  less. 

When  the  sluice  is  fully  drawn,  and  the  reservoir  is  wide,  wl  may 
be  assumed  =  0.  Hence,  in  Jonval's  turbine,  the  efficiency  decreases 
as  the  quantity  of  water  diminishes,  or  as  the  sluice  is  lowered.  In 
Fontaine's  turbines,  the  same  relative  effects  are  produced  for  dif- 
ferent positions  of  the  sluice  as  in  Fourneyron's  turbines.  It  appears, 
therefore,  that  the  efficiency  of  the  turbines  now  under  discussion, 
cannot  be  much  more  or  less  than  that  of  Fourneyron's  in  the  same 
circumstances.  Experiments,  hereafter  cited,  confirm  this. 

§169.  Construction  of  Fontaine's  andJonvaTs  Turbines.  —  We 
have  now  to  determine  the  general  rules  for  the  proportions  and 
construction  of  these  wheels. 

The  angles  /3  and  5  of  the  wheel-buckets  are  taken  arbitrarily  —  the 
latter,  however,  as  small  as  possible,  i.  e.,  15°  to  20°,  and  ft  =  100° 
to  110°.  From  these  we  have  the  guide-curve  angle  a,  if,  for  the 
sake  of  preventing  all  impact  at  entrance  of  the  water,  we  put 

'.  .  ,  c,  sin.  o 

c.  sin.  /3  =  <?2  sin.  8  =  v  sin.  8.  and  -i-  =  —  :  —  -  --  -. 

v       sin.  (ft  —  a) 

Hence,  by  combination  :  _  -—  _  =  *m'  S  ;  and  we  have 
sin.  (ft  —  o)       sin.  ft 


or 


, 

sin.  a  sin.  ft       sin.  8  *m.  8 

From  the  angles  a  and  p,  we  have  the  velocity  of  the  wheel 


V7' 


sin.  ft  cos.  a         /      sin.  ft 


,  a       „  /  _  sin,  ft      \ 
*'*'    \sin.   a  —  ft  / 


sin.  (ft  —  *)         \sin.  (a  —  ft) 


and  the  velocity  of  entrance  of  the  water  c  =  —  :  —  -  —  *  —  -  ;  and  from 

sin.  (ft  —  o) 

this  we  have  the  sectional  areas  F=  —  ,  and  F2=  — 

c  c 

The  width  of  the  wheel,  or  length  of  the  buckets  measured 
radially,  must  be  made  in  suitable  proportion  (as  small  as  possible), 
VOL.  II.  —  25 


290  CONSTRUCTION  OF  THE  BUCKETS. 

v  =  -  to  the  mean  radius  of  the  wheel.     In  the  turbines  hitherto 
r 

made  *  =  0,3  to  0,4.     This  being  done,  we  have  : 

JP  =  2  x  r  d  sin.  a.  =  2  ft  .  v  r2  sin.  a, 

J~       Kt 
,  and  d  =  v  r. 
2  rt  v  Sin.  a 

If  we  further  assume  a  proportion  4-  =  -  of  the  width  of  the  orifice 

d 

measured  at  the  mean  circumference  of  the  wheel  to  the  length  of 
the  buckets  (in  existing  turbines  this  is  J),  we  have  e  =  4  d,  and, 

hence,  the  number  of  buckets  n  =  2  *  r  sin'  a  =  ?-.     The  height  of 

e  de 

the  wheel  b  is  made  about  the  same  as  the  width  d, 

§  170.  Construction  of  the  Buckets. —  The  buckets  are  surfaces 

of  double  curvature,  the  generatrix  of  which  passes,  on  the  one  hand, 

through  the  axis  at  right  angles,  and  on  the  other  through  a  leading 
line  which  we  may  suppose  drawn  on  a 
Fig.  279.  cylinder  of  the  mean  radius  r.     As  by 

developing  a  cylinder  as  a  plane,  a 
rectangular  surface  is  produced,  lines 
may  be  drawn  on  this  surface,  which, 
when  the  right  angle  is  round  the  cylin- 
der, will  serve  as  the  leading  line  for  the 
bucket  surfaces.  These  developed  lead- 
ing lines  may,  however,  be  constructed 
of  arcs  and  their  tangents.  If  KL,  Fig. 
279,  be  the  developed  circle,  in  which 
the  wheel  and  guide -curve  apparatus 

meet,  the  line  AND  of  the  guide-curve,  may  be  found  by  making 

AA1  =  — —  ,  and  by  drawing  AN,  A^. . .  . ,  so  that  the  angle  of 
n 

inclination,  NAL  =  N^AJJ  . . .  =  a.  Again,  let  fall  AO  perpen- 
dicular to  A^.  Draw  a  line  parallel  to  KL  at  a  distance  equal 
to  the  height  of  the  guide-curves.  From  the  point  0  where  the 
perpendicular  AO  intersects  this  line,  describe  the  arc  NJ)V  and 
similarly,  from  another  point  0,  the  arc  ND,  &c.  &c.,  then  AND, 
A1N1D1  are  the  developed  guide  lines  of  the  guide-curves.  To 
find  the  guide  lines  of  the  wheel  buckets,  draw  at  the  distance 
EL  =  b  the  height  of  the  wheel,  the  line  EG  parallel  to  KL,  make 

EE,=  ?-fLT,  and  draw  the  straight  lines  EB,  JE^,  &c.,  so  that 

the  angle  BECr  =  B^ft  becomes  equal  to  the  angle  of  discharge  6. 
Again,  let  fall  EJ$  perpendicular  to  BE,  and  lay  off  AB,  so  that 

the  angle  ABC=  1±-^'.     If,  lastly,  from  the  centre  M  of  the  line 

AB,  there  be  raised  the  perpendicular  MC,  this  cuts  BT  at  the 
centre  0  of  the  arc  AB,  forming  the  upper  part  of  the  developed 


FONTAINE'S  AND  JONVAL'S  TURBINES.  291 


guide  line  of  a  wheel  bucket,  whilst  the  straight  lines  BE,  BVEV  &c., 
form  the  lower  part. 

It  is  evident  that  this  construction  of  the  guide  and  wheel  curves, 
insures  that  water  leaves  them  with  the  sections  A^  and  BE^  re- 
spectively. 

Example.  It  is  required  to  give  the  leading  dimensions  and  proportions  of  a  JonvaPs 
turbine  for  a  fall  of  12  feet  in  height,  with  8  cubic  feet  of  water  per  second.  Assuming 
J=20°,  and  8=  105°,  we  have  for  the  angle  of  the  guide  curves: 

cotg.  «  =  cotg.  8  +  -J—  =  cotg.  105°  +       1       —  _  0,26795  +  2,92380  =  2,65585, 

and,  therefore,  a.  =  20°,  38'.     Assuming  £  =  0,15,  and  it  =  0,10,  the  best  velocity  for  the 
wheel : 

2gh  8.02  </\2 


iin.ficos.a.   ^/ sm.Ji \  +  9       v/1,813  +  0,1407  +  0,1000 

27  78 
=  —    '          =  19,38  feet,  and  from  this  we  have  the  velocity  of  entrance  : 

1/2,0531 

vnn.8    _  1S7?  f 
sin.  (8  —  *) 

The  sections  F  =  SL  =  _JL  =  0,4262  square  feet,  and  F2  =-^  =  _8_  =  0,4127 
c        ]  8,77  v        19,38 

square  feet,  and  if  we  take  the  ratio  v  =  —  =  5,  the  mean  radius : 

r  =     I F        =    / °'4'26- =  0,7598  feet,  and  the  width  of  wheel 

*J  2  TT  v  sin.  a       *Jf  ir  sin.  20°,  38' 

d  =  v  .  r  =  °'7598  =  0,2532  feet.    From  the  space  occupied  by  the  buckets,  each  of 

3 
these  calculated  dimensions  should  be  somewhat  increased. 

The  width  of  the  channels  e  =  £  d  =  0,1266  feet,  and  .-.  n  the  number  of  buckets 
_f^_          0,4262          _  °'4262  _  13,29,  for  which  we  may,  however,  adopt  16. 

de       0,2532  .  0,1266         0,032 

The  height  of  the  wheel  b  is  made  =  d  =  0,2  532.     The  radius  of  the  reservoir  may 

be  made  somewhat  greater  than  r  +  d-  =  0,7598  +  0,1266  =  0,8864,  or  about  1  foot, 
and  hence  the  area  of  it  will  be  it  =  3,1416  square  feet.  The  velocity: 

u,  __  Q  =       8      —  2,546  feet,  and  the  height  due  to  this  velocity: 

ir       3,1416 

2  =  0,0155  .  2,546* =0,1 005  feet.  The  effect  of  this  wheel,  when  the  sluice  is  com- 
pletely drawn,  would  be : 

—  (12  —  TO  15  18,773+  0,10 . 19,38"  +  (2  . 19,38  sin.  10°)*  +  2,546'] .  0,0155)  8  .  62,5 
=  (12  —  103,66  .  0,0155)  .  500=10,39  X  500=  5195  ft.  Ibs.,  =  9,4  horse  power. 
The  losses  of  effect  in  the  reservoir  would  reduce  this  to  4800  ft.  Ibs.,  so  that  the  efficiency 
would  be  something  near  0,80;  for  as  the  power  expended  is  62,5  X  8  X  12  =  6000  ft. 
Ibs.,  4800  -i-  6000  =  ,80.  These  calculations  are  for  English  measures. 

§  171.  Experiments  on  Fontaine's  and  JonvaVs  Turbines.—Verj 
trustworthy  experiments  on  these  wheels  are  detailed  in  the 
"  Comptes  Rendues  de  1'Acade'mie  des  Sciences  a  Paris,  1846.' 
There  are  also  some  earlier  experiments  by  MM.  Alcan  and 
Grouvelle.  (See  Bulletin  de  la  Socie'te'  d'Encouragement,  tome  xliv.  ) 

These  experiments  show  that  in  Fontaine's  turbines,  as  in  1  our- 
neyron's,  the  efficiency  is  greatest  when  the  sluice  is  quite  drawn 
up,  and  that  the  efficiency  is  less  affected  by  variations  of  head, 


292  FONTAINE'S  AND  JONVAL'S  TURBINES. 

than  by  variations  in  the  quantity  of  water  supplied.  The  turbine 
at  Vadeney,  near  Chalons-sur-Marne,  the  efficiency  of  which  was 
determined  by  Alcan  and  Grouvelle,  has  1,6  metres  (5,24  feet)  ex- 
ternal diameter,  0,12  metres  (nearly  5  inches)  in  height,  the  fall 
was  5J  feet,  the  quantity  of  water  about  93  gallons  per  second. 
The  principal  result  of  this  experiment  was  that  for  u  =  30  to  50 
per  minute,  the  efficiency  was  0,67.  One  of  Fourneyron's  wheels 
\  of  an  early  date,  made  for  the  same  fall,  gave  7  =  0,60.  Morin's 
experiments  were  made  on  a  turbine  for  a  powder  mill,  at  Bouchet. 
The  diameter  was  1,2  metres,  the  width  0,25  metres.  There  were 
24  guide-curves  and  58  wheel  buckets.  It  had  a  fall  of  about  12 
metres,  and  6  cubic  feet  per  second  supply.  Experiments  were 
made  with  2,  3,  and  4  inches  of  the  sluice  drawn,  and  the  following 
results  obtained.  Sluice  quite  open  u  =  45,  the  efficiency  a  maxi- 
-  mum,  and  =  0,69  to  0,70. 

When  the  sluice  was  shut  so  as  to  reduce  the  expenditure  by  J, 
17  was  reduced  to  0,57.  The  efficiency  varied  little  with  the  velocity 
of  the  wheel,  for  when  making  35  revolutions  per  minute,  17  was  still" 
=  0,64,  and  for  55  revolutions,  q  =  66.  It  appears,  too,  that  the 
greatest  power  exerted,  and  at  which  the  wheel  moved  irregularly, 
was  about  1^  times  that  with  which  the  wheel  produced  its  maximum 
effect.  The  wheel  was  a  few  inches  in  back  water  during  the  ex- 
periments. We  see  from  these  experiments,  that  Fontaine's  turbine 
may  be  considered  among  the  first-class  of  hydraulic  wheels.  The 
circumstance  of  the  pivot  being  out  of  water  is  an  advantage  (though 
obtained  at  considerable  expense,  and  by  a  method  inapplicable  to 
large  machines).  The  "graissage  atmosphe'rique"  of  Decker  and 
Laurent  accomplishes  the  same  end,  the  lower  end  of  the  upright 
shaft  being  surrounded  by  a  bell,  analogous  to  a  diving-bell,  which 
revolves  with  it.  The  air  in  the  bell  is  kept  of  the  necessary  den- 
sity by  a  small  air-pump. 

§  172.  JonvaVs  Turbines.  —  The  experiments  on  Jonval's  tur- 
bines gave  equally  favorable  results  as  those  on  Fontaine's.  Messrs. 
Kochlin  and  Co.  have  detailed  experiments  on  one  constructed  by 
them  at  Muhlhausen,  in  the  "Bulletin  de  la  Socie'te'  Industr.  de 
Mulhause,  1844."  This  turbine  was  3,1  feet  in  diameter,  8  inches 
high.  It  was  placed  2;  —  8"  under  the  surface  of  the  water  in  the 
lead,  the  fall  being,  however,  5J  feet,  and  the  supply  being  125 
gallons  per  second.  The  efficiency  for  u  =  73  to  95  per  minute 
was  0,75  to  0,90.  Morin  considers,  however,  that  the  quantity  of 
supply  was  reckoned  too  low,  and  that,  therefore,  this  high  efficiency 
must  be  reduced  from  0,63  to  0,71. 

Colonel  Morin  made  experiments  with  a  turbine  of  0,81  metres 
external  diameter,  0,12  metres  internal  width,  18  buckets,  fall  5J 
feet,  supply  45  to  65  gallons  per  second.  Morin  comes  to  the  fol- 
lowing conclusions  from  all  his  experiments.  In  the  normal  state, 
the  water  having  impeded  entrance  and  exit,  the  number  of  revolu- 
tions was  90  per  minute  and  rt  =  0,72.  By  putting  contracting 


COMPARISON  OF  TURBINES.  293 

pieces  on  the  wheel,  the  efficiency  did  not  become  much  less  (0,63) 
until  the  section  was  very  considerably  diminished. 

The  efficiency  did  not  vary  for  variations  of  velocity  25  per  cent, 
above  and  below  that  for  the  maximum  effect.  By  depressing  the 
sluice,  the  efficiency  was  diminished,  so  that  it  is  evidently  a  very  im- 
perfect regulator  for  the  wheel.  When  the  section  of  the  aperture 
for  the  discharge  of  the  water  was  reduced  to  0,4  of  that  for  the 
normal  condition,  n  was  reduced  to  0,625. 

Redtenbacher  gives  some  experiments  on  a  turbine  of  Jonval's, 
the  maximum  efficiency  for  the  sluice  fully  drawn  having  been 
=  0,62.  As  in  the  case  of  Fourneyron's  turbines,  these  experi- 
ments indicate  that  the  wheel  working  without  load  makes  about 
twice  as  many  revolutions  as  when  furnishing  its  maximum  effect  in 
its  normal  state. 

§  173.  Comparison  of  different  Turbines  with  each  other. — If  we 
compare  the  turbines  of  Fontaine  and  Jonval  with  those  of  Four- 
neyron,  we  find  that  in  Fontaine's  turbines  the  water  is  less  deviated 
from  its  original  direction  of  motion  than  in  Fourneyron's,  so  that 
for  the  same  velocity  of  entrance  the  resistance  is  less  in  the  one 
than  in  the  other.  Thus  the  velocity  of  entrance  in  Fontaine's 
wheel  may  be  made  greater,  and,  therefore,  the  wheel  may  be  made 
less  in  diameter  than  Fourneyron's.  The  "guide-curves  of  Fontaine's 
wheels  take  on  the  water  in  more  nearly  parallel  layers  than  they 
do  in  Fourneyron's  wheels,  where  a  divergence  of  the  stream  enter- 
ing the  wheel  cannot  possibly  be  avoided. 

On  the  other  hand,  Fourneyron's  wheels  have  certain  advantages. 
The  pressure  on  the  pivot  is  reduced  to  the  weight  of  the  machine 
in  motion  ;  whilst  in  Fontaine's,  the  whole  weight  of  water  is  borne 
by  the  pivot,  thus  involving  greater  friction,  caeteris  paribus.  Again, 
in  Fourneyron's  turbines,  the  particles  of  water  move  with  the  same 
velocity  of  rotation,  which  is  not  the  case  in  the  newer  turbine,  in 
which  the  velocity  of  the  outer  particles  is  much  greater  than  that 
of  the  inner.  This  givet  rise  to  eddying  motions,  consuming  me- 
chanical effect,  and  causing  irregularities  in  the  motion  of  the  water 
through  the  wheel.  The  turbine  of  Fourneyron  is  also  more  easily 
constructed  than  that  of  Fontaine,  particularly  the  buckets. 

Remark  1.  The  Fontaine  turbines  are  well  adapted  for  tide-mills. 

Remark  2.  Jonval's  turbines  are  considered  to  present  advantages  in  respect  of  their 
being  placed  so  that  they  can  be  easily  got  at.  The  limit  at  which  they  may  be  placed 
above  the  tail-race  has  been  already  pointed  out  to  be  34  feet;  but  from  experiments  of 
M.  Marazeau,  and  from  certain  theoretical  considerations  of  Morin,  it  appears  that  the 
height  of  the  turbine  above  the  water  in  the  race  must  not  exceed  even  lower  limits 
than  the  above,  because  otherwise,  the  water  is  very  apt  to  lose  its  continuity  imme- 
diately under  the  wheel,  and  thus  effect  is  lost. 

§  174.  Comparison  between  Turbines  and  other  Water  Wheels.— 
Turbines,  from  their  nature,  are  applicable  to  falls  of  any  height, 
from  1  to  500  feet.  Vertical  water  wheels  are  limited  in  their 
applications  to  falls  under  60  feet  as  the  highest.  The  efficiency  of 
turbines  for  very  high  falls  is  less  than  for  smaller  falls,  on  account 
of  the  hydraulic  resistances  involved,  and  which  increase  as  the 

25* 


294  COMPARISON  OF  TURBINES. 

square  of  the  velocity.  Vertical  water  wheels  having  from  20  to  40 
feet  fall,  give  a  greater  efficiency  than  any  turbine.  For  falls  of 
from  10  to  20  feet,  they  may  be  considered  as  being  very  nearly  on 
a  par  in  point  of  efficiency ;  and,  for  very  low  falls,  turbines  give  a 
higher  efficiency  than  any  vertical  wheel  that  could  be  substituted 
for  them.  Poncelet's  wheels,  for  falls  of  from  3  to  6  feet,  are  on  a 
par  with  turbines,  but  only  within  these  limits.  Turbines  are  un- 
affected by  back-water,  whilst  vertical  wheels  lose  effect  in  this 
condition.  Variations  in  supply  of  water  affect  the  efficiency  of 
vertical  water  wheels  less  than  they  do  that  of  turbines.  This 
gives  the  vertical  water  wheel  an  hydraulic  economical  advantage, 
which  is  in  some  cases  of  great  importance.  When  water  becomes 
scarce,  the  best  effect  from  what  is  available  may  always  be  de- 
pended upon  from  a  good  vertical  wheel,  whilst  the  turbine  falls  off 
in  efficiency  as  its  sluice  is  lowered,  from  causes  which  in  our  dis- 
cussion of  the  theory  of  turbines  we  have  fully  explained. 

§  175.  Variations  of  velocity  on  either  side  of  the  normal  con- 
ditions, have  the  same  result  in  the  two  kinds  of  wheels,  but  the 
turbines  have  a  decided  advantage,  in  that  they  make  a  greater 
number  of  revolutions  per  minute  than  any  vertical  wheels.  The 
velocity  of  rotation  is  limited  to  from  4  to  8  feet  per  second, 
whilst  in  turbines  this  velocity,  having  a  certain  ratio  to  the  height 
of  fall,  is  generally  much  greater.  The  application  of  water  to  ope- 
.  rations  requiring  great  velocity,  is,  therefore,  most  advantageously 
made  by  turbines ;  whilst  for  operations  requiring  slow  motions,  the 
vertical  wheel  is  to  be  preferred.  It  is  a  question  of  practical  dis- 
cretion, to  decide  as  to  whether  it  is  better  to  reduce  the  velocity 
of  turbines,  or  to  raise  the  velocity  of  vertical  wheels  by  means  of 
the  gearing  that  is  to  transmit  their  water-power  to  the  work  to  be 
done. 

For  variable  resistances,  such  as  rolling  mills,  forge  hammers, 
&c.,  the  vertical  wheel  is  certainly  to  be  preferred,  because  its  great 
mass  serves  better  for  regulating  the  motten  than  the  smaller  tur- 
bine, which  for  such  work  requires  the  addition  of  a  fly-wheel. 

In  respect  to  economy  of  construction,  turbines  are  at  least  as 
cheap  as  vertical  wheels.  When  the  fall  is  considerable  and  the 
quantity  of  water  great,  the  turbine  is  the  cheaper  machine  of  the 
two.  The  turbine  almost  necessarily  involves  the  use  of  iron  in  its 
construction,  and  hence  cannot  always  be  adopted.  The  durability, 
or  the  maintenance  of  a  turbine,  is  probably  less  than  that  of  a 
vertical  wheel,  cseteris  paribus. 

In  respect  to  workmanship,  it  is  manifest  that  the  guide-curve 
turbines  require  greater  skill  than  vertical  water  wheels  do  for  their 
construction,  with  the  same  relative  degree  of  perfection.  Also, 
deviation  from  the  scientific  rules  for  their  construction  is  of  much 
more  prejudicial  consequence  for  turbines,  than  in  the  case  of  vertical 
wheels.  This  latter  circumstance  is  the  cause  of  the  failure  of  many 
of  the  turbines  that  have  been  erected,  and  operates  against  their 
more  general  introduction. 


TURBINES  WITH  HORIZONTAL  AXIS. 


295 


Turbines,  it  must  be  borne  in  mind,  require  clean  water  to  be  laid 
on,  for  they  would  be  greatly  damaged  by  sand,  mud,  leaves,  branches, 
ice,  &c.,  passing  through  them,  and  their  efficiency  lessened.  This 
is  not  the  case  with  vertical  wheels. 

§  176.  Turbines  with  Horizontal  Axis. — Examples  of  distorted 
ingenuity  have  been  displayed  in  putting  turbines,  particularly  Jon- 
val's  and  Whitelaws',  on  horizontal  axes.  This  mode  of  construc- 
tion can  never  be  advantageous,  though  it  may  have  some  local 
convenience  suggesting  its  adoption. 

Jonval  and  Redtenbacher  have  proposed  the  arrangement  shown 
in  Fig.  280.  Where  AA  is  the  lead  pipe,  BB  the  one,  and  £1fi1 

Fig.  280. 


Fisr.281. 


the  other  wheel,  CC^  the  horizontal  axis,  and  DD  and  DDl  the 
jointing-rings  (Vol.  II.  §  151),  E  and  El  being  the  tail-race. 

A  throttle  valve  in  the  main  or  lead  pipe  is  the  means  of  regulation. 
Herr  Schwamkrug,  of  Freyberg,  has  recently  erected  a  vertical 
wheel,  working  on  the  principle  of 
the  pressure  turbine.  The  wheel  is 
like  one  of  Poncelet's,  but  the  water 
is  introduced  on  the  inside  by  a  pipe, 
so  that  it  flows  through  the  wheel 
near  the  bottom  of  it.  Fig.  281  shows 
the  arrangement  adopted. 

The  guide-curves  DE,  D,E,  are 
movable  on  centres,  and  serve  to  regu- 
late the  discharge  of  water.  This 
construction  has  advantages  in  respect 
of  the  wheel  being  little  exposed  to 
and  as  the 


the  action  of  the  water, 
water  acts  on  a  very  small  arc,  the  wheel  must  have  a  greater 
diameter  than  a  turbine,  and  hence  in  cases  where  slow  motion  is 


296 


TURBINES  WITH  HORIZONTAL  AXIS. 


required,  may  do  away  with  the  necessity  of  intermediate  gear  for 
reducing  speed.  But  such  a  wheel  would  necessarily  be  more  costly 
than  a  turbine,  and  its  efficiency  would  certainly  be  less. 

The  same  principle  might  be  applied,  as  shown  in  elevation  in 
Fig.  282,  to  a  Fontaine's  turbine.  Such  a  machine  is  applicable  for 
all  falls,  but  never  advantageously. 

Before  concluding  this  subject,  we  may  add  that  Poncelet's  tur- 
bines have  been  quite  recently  applied  in  Switzerland,  under  the 
name  of  tangential  wheels. 


Fiz.  282. 


Fig.  283. 


Fig.  283  represents  a  horizontal  section  of  a  part  of  one  of  these 
wheels,  and  the  mode  of  laying  on  the  water.  S  is  the  regulating 
sluice,  in  advance  of  which  the  lead  is  divided  into  three  channels 
by  guide  plates.  The  water  is  discharged  in  the  interior  of  the 
wheel  in  such  manner  that  the  pivot  is  protected  from  the  water. 

[Mr.  Ell  wood  Morrh,  in  the  "Journal  of  the  Franklin  Institute,"  for  November,  1842 
(third  series,  vol.  iv.,  p.  303),  in  discussing  the  advantages  of  Fourneyron's  turbines, 
makes  the  following  remarks:  "  In  conclusion,  the  chief  points  of  advantage  promised 
by  the  use  of  turbines  upon  the  mill  seats  of  the  United  States,  may  be  briefly  summed 
up  as  follows: — 

1.  They  act  with  perfect  success  in  back-water. 

2.  They  are  not  liable  to  obstruction  from  ice.  .     • 

3.  They  require  but  little  gearing  to  get  up  a  high  velocity  at  the  working  point. 

4.  They  use  to  advantage  every  inch  of  fall. 

5.  They  are  equally  applicable  to  very  high  and  very  low  falls. 

6.  They  are  equal  in  power  to  the  best  overshot  wheels. 

7.  They  may  vary  greatly  in  velocity  without  losing  power. 

8.  They  are  very  compact  and  occupy  but  little  room. 

9.  They  may  be  very  accurately  regulated  to  an  uniform  speed. 

10.  They  are  perfectly  simple,  and  not  likely  to  get  out  of  order. 

11.  They  are  not  very  expensive. 

12.  They  are  very  durable. 

"Upon  one  account  or  another,'' he  adds,  " the  turbine  is  superior  to  all  other  water 
wheels,  and  consequently  must  be  regarded  as  the  very  best  hydraulic  motors  now  known 
to  mechanics." — AM.  ED.] 

Literature.  Tiie  literature  on  turbines  has  of  late  years  become  very  extensive.     We 


TURBINES  WITH  HORIZONTAL  AXIS.  297 

have  already  mentioned  several  treatises  and  papers  on  the  subject.  The  following  are 
some  of  the  more  important  works  : — 

Fourneyron's  original  paper  appeared  in  the  "  Bulletin  de  la  Societe  d'Encouragement, 
1834."  Morin's  "Experimental  Inquiry,"  already  quoted,  followed  in  1838.  In  1838, 
Poncelet  published  his  "  Theorie  des  Effets  mecaniques  de  la  Turbine  Fourneyron,"  in 
the"Comptes  Rendues,"  and  as  a  separate  treatise.  In  D'Aubuisson's  "  Hydraulique,1' 
the  turbine  is  treated  of,  but  only  superficially.  In  1843,  Combes  published,  "  Recherches 
theoretiques  et  experimentales  sur  les  Roues  a  reaction  ou  a  tuyaux,"  a  tract  of  con- 
siderable importance,  as  it  for  the  first  time  recognizes  the  necessity  of  taking  into  con- 
sideration the  hydraulic  resistances,  which  Poncelet  and  Redtenbacher  have  neglected 
to  do.  Redtenbacher's  work,  "  Theorie  und  Bander  Turbinen  und  Ventilatoren,  Man- 
heim,  1844,"  is  founded  on  Poncelet's  theory,  and  is  the  best  and  most  complete  work 
on  the  subject.  On  the  newer  turbines,  there  appears  in  the  "Comptes  Rendues,"  tome 
xxii..  1846,  "Rapport  sur  un  Memoire  de  M.  M.  A.  Koechlin,  concernant  une  nouvelle, 
Turbine  (Jonval)  construite  dans  leurs  ateliers,  par  Poncelet,  Piobert,  et  Morin."  Also, 
"  Note  sur  la  Theorie  de  la  Turbine  de  Koechlin.  par  Morin,"  et  "  Note  sur  1'Application 
de  la  Theorie  du  Mouvement  des  Fluides  anx  experiences  de  M.  Marozeau,  par  Morin.'' 
In  the  "Comptes  Rendues,"  &c.,  t.  xxiii.,  1846,  there  appears  a  paper  "  Experiences  et 
Notes  sur  la  Turbine  de  M.  Fontaine-Baron,  par  Morin."  The  "  Bulletin  de  la  Societe 
d'Encouragement,  1844-45,"  contains  notices  of  the  turbines  of  Jonval  and  Fontaine. 

Armengaud's  publication  "Industrielle,"  contains  good  drawings  and  descriptions  of 
the  turbines  of  Cadiat,  Callon.  Fourneyron,  and  Gentilhomme.  In  the  "  Polytech.  Cen- 
tra Iblatt,  bd.  vii.,  1846,"  Parro's  turbine  is  described.  Nagel's  turbine  is  described  in 
Dingler's  "  Journal,  bd.  xcv.,"  and  Passot's  turbine,  in  the  same  Journal,  bd.  xciv.  Bour- 
geois' screw,  is  a  turbine-helice,  or  with  screw-formed  channels.  See  "  Polytechnisches 
Centralblatt,  bd.  i.,  1847."  In  the  "  Proceedings  of  the  Institution  of  Civil  Engineers  for 
1 842,"  there  is  a  notice  of  turbines  by  Prof.  Gordon.  In  the  "  Transactions  of  the  Society 
of  Arts  of  Scotland,  1805,"  there  is  a  notice  of  a  turbine  erected  at  Mr.  J.  G.  Stuart's  flax- 
mill,  at  Balgonie,  in  Fifeshire.  This  is  the  first  turbine  erected  in  Britain,  and  is  one  of 
the  largest  ever  made.  Its  efficiency  is  reckoned  to  be  =  0,70. 


298 


WATER-PRESSURE  ENGINES. 


CHAPTER  VI. 


WATER-PRESSURE   ENGINES. 

§  177.     Water-Pressure  Engines. — Water-pressure  engines,  as 
their  name  indicates,  are  set  in  motion  by  a  column  of  water.     Their 
motion  is  a  reciprocating   rectilinear 
Fit?  284.  motion,  and  not  rotatory  as  in  the  tur- 

bine. The  leading  features  of  a  water- 
pressure  engine  are  delineated  in  Fig. 
284.  Ji  is  a  reservoir  at  the  upper 
end  of  the  pipe.  AR  is  the  pressure 
pipe.  C  is  the  working  cylinder,  in 
which  the  water  moves  the  loaded 
piston  K.  In  the  pipe  BC,  by  which 
the  pressure  pipe  communicates  with 
the  cylinder,  the  regulating  valve  or 
cock  is  placed.  It  is  here  represented 
as  a  three-way  cock,  serving  alternately 
to  open  and  close  the  communication 
between  the  working  cylinder  and  the 
pressure  pipe.  When  the  way  is  open, 
the  water  presses  on  the  piston,  and 
raises  it,  with  its  load,  through  a  cer- 
tain height — the  length  of  stroke  — 
when  the  communication  between  the 
pressure  pipe  and  the  cylinder  is  shut,  a 
way  is  opened  for  the  discharge  of  the 
water  from  the  cylinder  by  the  pipe  J), 
and  the  piston  then  descends  by  its  own  gravity. 

Water-pressure  engines  are  either  single  or  double  acting.  Fig. 
284  shows  the  general  arrangement  of  the  single-acting  engine,  in 
which  the  piston  is  made  to  move  in  one  direction  by  the  pressure 
of  the  water,  and  to  return  by  its  own  weight. 

In  the  double-acting  engine,  the  up  stroke  and  down  stroke,  or 
both  strokes  of  the  piston,  are  made  under  the  hydraulic  pressure. 
Fig.  285  shows  the  general  arrangement  of  a  double-acting  engine. 
The  cock  is  in  this  case  a  four-way  cock.  In  I.  the  pressure  is  on 
the  upper  side  of  the  piston  through  JlBC,  and  the  discharge  goes 
on  through  C^J).  In  II.  the  pressure  is  on  the  under  side  through 
AB^Cl  of  the  piston,  and  the  discharge  through  CBD. 

Water-pressure  engines  are  also  made  with  two  cylinders,  each 
single-acting,  but  connected  together,  as  in  Fig.  285,  so  that  while 
the  one  piston  is  ascending  by  the  pressure  of  the  water,  the  other 
is  descending,  the  water  being  discharged  therefrom.  The  relative 


PRESSURE  PIPES. 


299 


oi  SLthe  PassaSes  in  the  four-way  cock  are  shown  in  Figs 
zoo  and  287. 

Fig.  285. 


§  178.  Pressure  Pipes.  —  The  pressure  pipes  should  take  the 
water  from  a  feeding  cistern  or  settling  reservoir,  in  which  the  water 
has  time  to  deposit  the  foreign  matters  it  may  have  carried  so  far 
along  with  it.  In  front  of  this  a  grating  must  be  placed,  to  keep 
back  leaves,  ice,  &c.  &c. 

The  end  of  the  pressure  pipes  should  dip  so  as  to  be  1|  foot,  at 
least,  above  the  bottom  of  the  feed  cistern,  and  3  to  4  feet  under 
the  surface  of  the  water  in  it,  so  as  to  prevent  the  influx  of  heavier 
particles,  and  to  render  the  indraught  of  air  impossible.  For  this 
object  the  end  of  the  pipe  may  be  conveniently  curved  with  the 
mouth  downwards,  as  shown  in  Fig.  288.  0  being  a  valve  for 
shutting  off  the  water  from  the  pipe  -B,  when  required.  F  is  a 
division  plate  in  the  cistern.  Cr  is  a  grating  to  keep  back  floating 
bodies.  The  pressure  pipes  may  be  either  of  wood  or  iron,  but  are 
usually  of  the  latter  material,  and  made  from  ^  to  ^  the  internal 
diameter  of  the  working  cylinder.  The  pipes  for  great  heads  are 
made  to  increase  in  thickness  from  the  top  downwards  proportionally 
to  the  pressure.  The  formula:  e  =  0,0025  n  d,  +  0,66  inches  may 
be  used  for  calculating  the  strength  required  for  any  given  pressure 
n  in  atmospheres  =  33  feet  of  water;  d,  being  the  internal  diameter 
of  the  pipes.  The  formula  given  in  Vol.  I.  §  283,  is  applicable  to 


300 


THE  WORKING  CYLINDER. 


ordinary  water  conduits,  but  is  inapplicable  to  the  present  case, 
because  the  pressure  of  the  water  here  varies  frequently,  and  even 
acts  with  impact,  when  the  valves  are  suddenly  closed.  The  pipes 

Fig.  2S8. 


must  be  carefully  proved  by  an  hydraulic  or  Bramah  press.  The 
porosity  of  pipes,  which  at  first  proving  is  very  sensible,  gradually 
becomes  insensible  as  oxidation  goes  on.  In  the  case  of  the  pipes 
for  the  pressure  engine,  at  .Huelgoat,  described  hereafter,  boiled  oil 
was  used  in  proving  the  pipes,  by  which  they  become  impregnated 
to  a  certain  depth  with  the  oil,  and  thus  their  porosity  stopped,  and 
even  protection  against  corrosion  insured. 

The  pressure  pipes  are  usually  jointed  by  flanches  and  screw- 
bolts;  a  ring  of  lead,  or  of  iron  rust  being  interposed,  as  shown  in 
Pigs.  289  and  290.  A  mixture  of  lime  water,  linseed  oil,  varnish, 


Fig.  289. 


Fig.  290. 


and  chopped  flax,  makes  a  very  good  pipe-joint.  The  spigot  and 
faucet  joint,  with  folding  wedges  of  wood,  make  the  best  and  cheapest 
joint  for  cast  iron  pipes. 

§  179.  The  Working  Cylinder. — The  working  cylinder  is  made 
of  cast  iron  or  of  gun  metal.  The  number  of  strokes  is  limited  to 
from  3  to  6  per  minute,  so  that  there  may  be  the  least  possible  loss 
of  effect;  and,  therefore,  the  capacity  of  the  cylinder  is  made  to 
depend  rather  on  its  length  than  its  diameter.  The  stroke  s  is  made 
from  3  to  6  times  the  diameter  d  of  the  cylinder.  The  mean  velo- 
city v,  of  the  piston,  is  usually  1  foot  per  second,  in  order  that  the 


THE  WORKING  CYLINDER.  301 

mean  velocity  v,  of  the  water  in  the  pressure  pipes,  and  hence  the 
hydraulic  resistances  may  be  as  small  as  possible.  It  is  not  advisa- 
ble in  any  case  to  have  the  latter  velocity  greater  than  10  feet  per 
second,  and  6  feet  is  a  better  limit.  If  we  assume  v  =  1,  and  v  =  6 
feet,  the  quantity  of  water  being :  ^  =  *  d^  l\  we  get  for  the 
proportion  ofjhe  diameter  of  the  pressure  pipe  to  that  of  the  cylin- 
der, i  =  IL  =  1 1  0,408,  or  about  0,4. 
«  \  vl  \  b 

If  Q  be  the  quantity  of  water  supplied,  per  second,  then  for  a 
double-acting  engine,  or  for  a  double-cylinder  engine,  Q  =  'L^.  .  Vf 
and  hencejve  have  the^  diameter  of  the  working  cylinder  required 
d  =  ft-?  =  1,13  15,  that  is,  for  v  =  1,  d  =  1,13  v/0"feet.  For 

\  rt  v  \V 

a  single-cylinder,  single-acting  engine,  Q=  |  .  £ —  v  .-.  d  =  1,60 


J 


— ,  and  if  v  =  1,  d  =  l,60v/Q  feet.  If  the  stroke  of  the  piston 
=  3  d  to  6  d,  the  time  for  one  stroke  of  a  single-acting  engine  is 
t  =  -,  or,  if  v  =  1,  t  =  s  in  seconds,  and  hence  the  number  of  sin- 
gle strokes  per  minute: 

60"       60  .  v         .  ,  60 

nl  = = .•.  when  v  =  1,  wt  =  — , 

and  the  number  of  double  strokes  : 

n.       30  t;        .,.         ..  30 

n  =  -i  = ,  or  if  v  =  1,  n  =  — 

2  8  S 

It  is,  however,  better,  in  the  case  of  a  single  acting,  single  cylinder, 
water-pressure  engine,  to  begin  the  stroke  somewhat  more  slowly, 
or  to  cause  the  descent  of  the  piston  to  take  place  more  rapidly  than 
with  the  mean  velocity,  because  the  hydraulic  resistances  are  greater 
for  the  working  or  up  stroke,  than  for  the  return  of  the  piston. 

The  working  cylinder  must  be  accurately  bored.  The  thickness 
of  the  metal  is  made  greater  than  the  usual  rules  of  calculation  indi- 
cate as  enough,  to  compensate  for  wear,  and  because  of  the  shock 
at  entrance  of  the  water.  The  formula  e  =  0,0025  n  d  +  1  will  be 
found  useful  in  guiding  to  the  proper  dimensions.  The  cylinder 
may  be  strengthened  by  mouldings  or  ribs  cast  round  it. 

The  working  cylinder  is  subject  to  a  pressure  in  the  direction 
opposite  to  that  in  which  the  piston  moves,  equal  to  the  weight  of  a 
column  of  water  F  h  y,  F  being  the  area  of  the  base,  h  the  height, 
and  y  the  weight  of  a  cubic  unit ;  h  being  not  unfrequently  several 
hundred  feet,  this  pressure  of  the  water  is  very  considerable,  and, 
hence,  the  substructure  on  which  the  cylinder  rests  must  be  very 
strong.  Water-pressure  engines  are  erected  in  the  shafts  of  mines 
VOL.  n.— 26 


302 


THE  WORKING  PISTON. 


for  raising  water,  more  frequently  than  in  any  other  position,  and 
cannot,  therefore,  be  placed  on  the  solid  rock,  or  foundations  laid 
thereon,  but  have  to  be  supported  on  cross 
Fig.  291.  beams  or  arches  of  stone,  or  of  iron. 

Remark.  Besides  this  pressure,  the  cylinder  has  to  with- 
stand a  horizontal  pressure  in  the  direction  of  the  water 
entering  it,  and  proportional  to  its  section.  The  effect  of 
this  is  less  observable,  because  the  pressure  acts  at  a  point 
only  a  little  above  the  base  of  the  cylinder,  and  because 
the  pressure  pipe,  which  is  firmly  connected  with  the 
cylinder,  is  equally  pressed  in  the  opposite  direction.  In 
any  bend  or  knee  piece  JIB,  Fig.  291,  there  is  a  resultant 
pressure  CR  =  /?,  which  may  be  put 
=  P  v/2=  Ft  hy  .  y2,  Fl  being  the  area  of  the  pipe 
and  h  the  pressure  height. 

§  180.  The  Working  Piston.  —  The  main  piston  which  moves 
under  the  pressure  of  the  water,  consists  essentially  of  a  cylindrical 
disc  fitting  smoothly  into  the  cylinder.  To  make  this  piston  per- 
fectly tight,  and  at  the  same  time  not  to  cause  thereby  too  great  a 
resistance  to  motion,  a  packing  (Fr.  garniture;  Ger.  Liderung]  of 
hemp,  leather,  or  metal  is  applied,  either  on  the  piston,  or  in  the 
cylinder,  in  which  latter  case  the  piston  becomes  what  is  termed  a 
plunger  or  ram.  The  packing  of  the  pistons  of  water-pressure  en- 
gines is  usually  either  leather  or  metallic  rings.  They  are  adjusted 


Fig.  292. 


Fig.  293. 


THE  PISTON  ROD  AND  STUFFING  BOX.  3Q3 

to  a  pressure  proportional  to  the  column  of  water,  so  that,  on  the 
one  hand,  no  water  may  escape  or  pass,  and  on  the  other,  that  there 
may  be  no  unnecessary  friction.  The  hest  packing  that  can  be  em- 
ployed, is  that  in  which  the  water  itself  presses  the  leather  or  pack- 
ing against  the  surface  of  the  cylinder,  or  of  the  ram.  The  packing 
is  made  so  that  it  can  be  gradually  compressed  as  it  wears,  by  means 
of  a  ring  fitting  upon  it,  and  adjusted  by  screws.  Fig.  292  is  the 
piston  of  a  water-pressure  engine  at  Clausthal,  in  which  the  manner 
of  laying  in  the  packing  is  clearly  represented.  A  is  the  piston, 
properly  so  called,  and  BB  the  piston  rod,  a  a  and  b  b  are  the  pack- 
ing rings,  and  c  c  two  fine  channels  communicating  with  the  back 
of  the  packing  b  b.  Other  methods  of  packing  we  shall  describe 
here*after. 

For  the  plunger  or  ram,  or  Bramah  piston,  the  packing  may  like- 
wise be  kept  tight  hydrostatically.  A,  Fig.  293,  is  the  piston,  B 
the  cylinder,  0  the  pressure  pipe,  DD  the  packing  or  stuffing  box, 
screwed  on  to  the  piston ,  a  a  is  the  packing  ring,  and  b  b  the  five 
channels  of  communication.  This  manner  of 
keeping  the  packing  tight  is  more  applicable  to  Fi^-  294- 

the  case  of  a  stuffing  box,  than  to  the  ordinary 
piston. 

Remark.  The  compressed  ring  packing  is  also  applied  at  the 
compensation  joints,  which  must  be  introduced  in  the  length 
of  the  pressure  pipe.  Fig.  294  shows  such  a  pipe,  AA  being 
the  enlarged  end  of  one  pipe  B,  accurately  bored  out,  and  rest- 
ing on  supportsCC;  a  a  are  packing  rings  compressed  by  screws 
and  nuts  on  to  the  thickened  end  of  the  upper  pipe  D. 

§  181.  The  Piston  Rod  and  Stuffing  Box.— 
The  piston  rod  goes  either  upwards  or  downwards 
to  the  open  end,  or  through  the  cover  of  the 
cylinder.  In  the  first  case,  it  requires  very  little  special  arrange- 
ment, and  may  be,  in  fact  is,  frequently  made  of  wood.  In  the 
second  case,  it  must  go  through  a  stuffing  box,  must,  therefore,  be 
turned,  and  can  only  be  made  of  iron  or  gun  metal.  The  dimen- 
sions of  the  piston  rod  is  to  be  calculated  according  to  the  received 
theory  of  the  strength  of  materials.  If  d  be  the  diameter  of  the 
working  cylinder,  and  p  the  pressure  of  the  water,  on  each  square 

inch  of  the  piston,  the  force  P  = p;  and  if  d2be  the  diameter 

of  the  piston  rod,  and  K  the  modulus  of  strength  of  its  material, 

T  2 

then  its  strength  =  P  =  — — —  K,  and  by  equating  the  two  forces, 

we  have  :  dz  =  d    liL     K  is  to  be  taken  from  the  table  in  Vol.  I. 

h 

§  186,  and  p  is  given  by  the  formula  p  =  — *-. 

The  stuffing  box  (Fr.  boite  d  garniture;  Ger.  Stopfbiichse)  is  a 
box  placed  on  the  cylinder  cover,  so  lined  with  leather  or  hempen 


304 


THE  VALVES. 


Fis  205. 


rings,  that  the  piston  rod,  in  passing  through  it,  has  freedom  to  move, 
but  the  passage  is  rendered  water,  or  air,  or  steam  tight,  according 
to  circumstances.  For  water-pressure  engines,  a  leather  packing  is 
found  to  answer  best.  Fig.  295  shows  the  apparatus  in  question. 
AA  is  the  piston  rod,  BB  the  stuffing 
box,  BaC  its  packing,  DD  the  cover 
with  the  screws  for  compressing  the 
packing.  A  grease  cup  is  sunk  in  the 
cover  _Z>,  and  kept  filled  with  a  grease 
composed  of  6  parts  hog's  lard,  5  parts 
tallow,  and  1  part  palm  oil,  or  with  pure 
olive  oil,  or  neat's-foot  oil. 

In  the  engine  at  Clausthal,  oiling 
presses  are  applied,  having  a  small  pis- 
ton, worked  by  a  weight,  and  which 
forces  the  grease  into  the  packing 
through  a  fine  tube  communicating  with  the  channels  of  a  brass 
ring,  having  a  section  of  the  I  form,  and  round  which  the  packing  is 
lapped. 

§  182.  The  Valves. — The  valves  and  their  gear  are,  as  it  werer 
the  very  heart  of  the  water-pressure  engine,  for  it  is  by  them  the 
machine  is  made  continuously  self-acting.  The  valves  cover  and 
uncover  apertures  for  the  admission  and  discharge  of  the  water 
from  the  cylinder,  and  these  are  worked  so  as  to  open  and  shut  the 
apertures  alternately,  by  means  of  gear  connected  with  moving  parts 
of  the  engine,  so  that  the  engine  is  thereby  raade  self-acting.  The 
valves  are  either  cocks,  or  sliding  pistons.  The  latter  form  is  now 
generally  adopted. 

The  manner  of  applying  a  cock  as  a  valve  has  been  already  ex- 
plained, so  that  we  shall  now  only  further  describe  the  sliding  piston 
valves.  The  arrangement  of  piston  valves  for  a  single  acting,  single 
cylinder  engine  is  shown  in  Figs.  296  and  29T.  M  is  the  pressure 


Fig.  296. 


Fig.  297. 


pipe,  C  the  working  cylinder,  B  the  valve  cylinder,  A  the  discharge 
pipe,  K  the  piston  valve,  and  L  its  counter  piston,  which,  by  taking 
the  equal  and  opposite  pressure,  readers  the  movement  of  the  valves 


THE  VALVE  COCK.  305 

more  easy.  When,  as  in  Fig.  296,  K  is  lowered,  the  working  cylin- 
der and  pressure  pipes  are  in  communication,  and  when,  as  in  Fig. 
297,  K  is  raised  to  the  position  K^  the  communication  between  the 
pressure  pipes  and  cylinder  is  shut,  and  the  passage  for  discharge 
of  water  from  the  cylinder  is  open.  In  the  double-acting  engine,  or 
in  the  double-cylinder  engine,  the  slide  pistons  must  be  arranged  as 
in  Figs.  298  and  299.  E  is  the  pressure  pipe,  Q  the  pipe°going 

Fig.  298.  Fisi.  299. 


to  the  top,  and  Cl  that  going  to  the  bottom  of  the  working  cylinder 
(or  going  respectively  to  the  bottom  of  the  two  cylinders  in  the 
double-cylinder  engine).  A  is  the  discharge  pipe  for  the  water  sup- 
plied by  the  first,  and  Al  that  for  the  water  supplied  by  the  second. 
From  Fig.  298,  we  see  that,  when  the  slide  valve  is  up,  the  pressure 
pipe  is  in  communication  with  0,  and  the  discharge  made  through 
A,  and  when  the  slide  pistons  are  lowered,  as  in  Fig.  299,  the  com- 
munication is  open  to  Gv  and  the  used  water  discharged  from  0  by 
the  pipe  Ar 

§  183.  The  Valve  Cock. — The  cock  is  used  for  smaller  engines, 
as  shown  in  Fig.  300.  HE  is  the  cock,  BB  its  cover,  K  is  the 
squared  end  on  to  which  a  lever  for  turning  it  fits,  D  is  a  screw  for 
raising  or  lowering  the  cock  in  its  cover.  The  passages  of  the  cock 
are  made  so  as  to  suit  the  purposes  to  which  it  is  applied,  as  we 
have  explained  above. 

In  Fig.  300,  a  means  of  counteracting  the  effects  of  greater  pres- 
sure coming  on  one  side  of  the  cock  is  shown;  b  bt  are  two  cuts  on 
the  cock,  communicating  with  the  passage  a,  by  the  openings  c,  <?„ 
so  that  a  counter  pressure  is  obtained,  which,  by  proper  adjustment 
of  the  parts,  cut  "out  at  b  and  bv  balance  the  diagonal  pressure  in 
the  main  passage. 

In  order  to  equalize  the  wear  of  the  cocks  on  all  sides,  Mr.  Brer- 
del,  of  Freiberg,  introduced  the  method  of  turning  them  round  cor- 
tinuously  in  the  same  direction,  instead  of  turning  them  backwards 
and  forwards  through  only  90°.  We  shall  see  the  application  of 
this  valve  in  a  description  of  a  water-pressure  engine  erected  by  M. 
Brendrel,  in  the  sequel. 

26* 


806 


THE  SLIDE-PISTON  VALVE. 


§  184.   The  Slide-piston   Valve. — The  pistons  are  generally  made 
of  slips  of  leather,  placed  one  above  the  other,  and  closely  packed 


Fig.  300. 


Fig.  301. 


together,  as  we  have  mentioned  for  the  packing  of  the  stuffing  box 
in  §  181.  The  engine  at  Huelgoat,  was  originally  made  with  cylin- 
drical slide  valves  of  gun  metal.  These  lasted,  without  repair,  for 
seven  years;  but  in  1839,  the  valves  having  worn  loose,  a  depth  of 
5  inches,  consisting  of  24  discs,  or  rings  of  leather  pressed  together 
and  accurately  turned  down,  was  substituted.  Reichenbach  made 
the  cylinder  valves  of  tin,  and  the  engines  in  Bavaria,  in  most  recent 
times,  have  had  the  valves  made  by  a  combination  of  leather  and 
tin  rings. 

At  the  end  of  the  stroke  of  the  working  piston,  the  valve  piston 
.i.K  (Fig.  301)  rises,  gradually  shutting  off  the  water  from  the 
cylinder,  but  in  gradually  checking  the  flow  of  water  in  the  course 
EC,  the  piston  is  pressed  on  one  side,  and  this  gives  rise  to  a  very 
rapid  wear.  To  prevent  this,  the  end  of  the  pipe  CD  communicat- 
ing with  the  working  cylinder  is  carried  quite  round  the  valve  cylin- 
der, so  that  it  incloses  it,  and  the  water  then  presses  equally  on 
every  side  of  the  piston,  as  it  moves  up  and  down.  The  packing 


Fig.  302. 


Fig.  303. 


THE  VALVE  GEAR.  3Qj 

suffers  by  this  arrangement,  as  it  has  room  to  expand  at  this  point 
and  has  to  be  compressed  as  it  passes  into  the  cylinder  above  01' 
below  it.  On  this  account  the  sup- 
ply of  water  to  the  cylinder  is  carried 
through  a  series  of  openings,  as 
shown  in  the  horizontal  section  in 
Fig.  302.  The  objection  to  this 
arrangement  is,  that  it  increases  the 
hydraulic  resistances.  The  form  of 
the  valve  piston  K  is  of  great  im- 
portance. The  communication  be- 
tween 0  and  E  must  not  be  sud- 
denly opened  or  shut,  so  that  the  column  of  water,  in  motion,  may 
not  be  suddenly  brought  to  rest ;  for  this  acts  violently  on  the  engine, 
on  the  same  principle  as  is  more  fully  developed  in  the  so-called 
hydraulic  ram.  The  gradual  opening 
of  the  communication  may  be  managed 
by  giving  the  piston  a  particular  form. 
We  shall  hereafter  show  how  a  slow  mo- 
tion of  the  valve  piston  is  effected,  and 
in  the  mean  time  point  out,  that,  by 
giving  a  conical  shape  to  the  head,  or 
that  part  of  the  piston  which  begins 
the  closing  of  the  ports,  a  ring-formed 
opening  is  made  between  0  and  E, 
which  is  gradually  diminished  as  the 
piston  ascends,  until  it  is  finally  closed. 
Besides  this  arrangement,  the  top  of 
the  slide  piston  is  perforated  by  slits 
that  gradually  diminish,  but  leave  a 
narrow  communication  between  0  and 
E,  even  when  the  ring-formed  opening 
above  mentioned  is  quite  closed,  so  that 
the  passage  is  not  perfectly  closed  un- 
til the  slide-piston  stroke  is  completed. 
This  system  of  coning  out  the  top,  and 
perforating  the  upper  part  of  the  pis- 
ton proper,  is  applied  in  the  Clausthal 
water-pressure  engine. 

§  185.  The  Valve  Gear.— The  gear 
for  moving  the  valves  of  water-pressure 
engines  is  generally  complicated,  more 
so,  for  instance,  than  in  the  steam  en- 
gine, because  water  is  practically  an 
incompressible  fluid,  exerting  no  pres- 
sure when  cut  off  from  the  pressure  column.  When  the  piston  K, 
Fig.  303,  in  ascending,  cuts  off  the  pressure  column  from  the  work- 
ing cylinder  O,  then  either  the  motion  of  the  working  piston  ceases, 
or,  in  virtue  of  its  vis  viva,  it  moves  away  from  the  water  in  the 


308 


COUNTER-BALANCE  GEAR. 


cylinder,  as  this  has  no  expansive  capability.  But  this  formation 
of  a  vacuum  under  the  piston  must  be  carefully  avoided,  and,  there- 
fore, the  valve  piston  should  begin  to  rise,  while  the  main  piston 
stroke  is  still  unfinished,  and  thus  the  vis  viva  of  all  the  parts  con- 
nected with  it  is  gradually  destroyed  by  the  gradual  cutting  off  of 
the  pressure  column.  But  although  the  stroke  of  the  piston  is  com- 
pleted as  the  slide  valve  closes  the  communication,  the  motion  of  the 
slides  must  not  stop  here.  The  water  in  the  working  cylinder  must 
now  be  discharged.  The  valve  must  rise  somewhat  higher,  in  order 
to  open  the  orifice  of  discharge.  Hence  it  is  not  possible  to  work 
the  valve  gear  directly  from  the  moving  parts  of  the  engine,  for  then 
the  motion  of  both  would  cease  simultaneously.  Intermediate  gear 
must  be  introduced,  by  which  the  motion  of  the  valve  piston  is  con- 
tinued after  the  working  piston  has  come  to  rest.  This  gear  may 
be  worked  either  by  weights,  raised  by  the  piston  in  its  ascent,  and 
let  fall  at  a  particular  part  of  the  course,  or  by  springs,  bent  during 
the  motion  of  the  piston,  and  disengaged  at  the  end  of  the  stroke, 
or  by  a  subsidiary  engine  regulated  by  the  main  engine,  but  whose 
working  piston  moves  the  valves  of  the  main  engine.  The  gear  of 
water  engines  is,  therefore,  either  counter-balance  gear,  spring  gear, 
or  water-prissure  gear. 

§  186.  Counter-balance  Gear. — This  gear  was  the  first  employed, 
and  is  now  found  only  as  the  older  water-pressure  engines,  under  the 
name  of  fall  bob,  valve  hammer,  and  other  names.  The  principle  of 
the  different  systems  is  always  the  same.  They  are  essentially  a 
heavy  weight  raised  by  the  working  piston,  and  suddenly  let  go  to 

Fig.  304. 


COUNTER-BALANCE  GEAR. 


309 


work  the  cocks,  or  valves,  by  means  of  linked  levers.  We  shall  here 
describe  only  two  of  these  arrangements.  The  small  engine  in  the 
Pfingstwiese  mine,  near  Ems,  has  gear  connected  with  a  pendulum 
or  fall  bob,  moving  two  pistons  S  and  T,  lying  horizontally  under 
the  working  cylinder  JT,  Fig.  304.  The  pendulum  swings  on  an 

Fig.  305. 


310  THE  VALVE  HAMMER. 

axis  (7,  and  consists  of  a  heavy  bob  6r,  and  two  fork-like  springs  FD 
and  F^DV  carrying  a  cross  head  D£DV  having  a  projecting  piece 
J?,  in  the  centre,  passing  between  two  small  rollers  on  the  valve  rod. 
The  bob  is  raised  so  as  to  exceed  the  summit  of  its  arc,  by  means  of 
link  work  CHNMLR,  connected  with  the  ram  of  the  engine  at  R. 
Motion  is  not  communicated  from  the  axis  of  the  pendulum  (7,  but 
by  means  of  an  arm  (70,  on  a  separate  axis,  and  forming  a  single 
bent  lever  with  OJT,  and  which  pushes  out  the  springs  FD  and  FlDl 
alternately,  so  far  that  the  bob  Gr  is  brought  beyond  the  position  of 
stable  equilibrium,  and  in  its  fall  gives  the  valve  rod  the  requisite 
extra  push  to  right  or  left.  At  the  commencement  of  the  stroke  of 
the  working  piston,  the  whole  apparatus  has  necessarily  a  very  slow 
motion.  The  coming  into  play  of  the  arm  (70,  on  the  one  or  other 
spring,  should  only  take  place  when  the  stroke  is  nearly  completed, 
that,  as  the  valve  piston  gradually  advances,  the  retarded  motion  of 
the  working  piston  may  begin. 

It  is  easy  to  perceive  from  our  figure,  how  the  pressure  water  is 
introduced  into  the  cylinder,  and  discharged  from  it  at  the  end  of 
the  stroke.  When  the  piston  iS  is  in  the  orifice  A,  the  pressure 
water  from  E  enters  by  the  opposite  orifice  into  the  cylinder ;  but 
if  8  be  in  the  orifice  next  E,  so  that  the  orifice  A  is  open  to  the 
cylinder,  then*  the  water  that  has  raised  the  ram  discharges  into  the 
waste-course  at  A. 

Remark.  This  little  engine  has  60  feet  fall,  4  feet  stroke,  1$  foot  diameter  working 
cylinder,  and  made  (in  1839)  1  stroke  in  65  seconds. 

§  187.  The  Valve  Hammer. — The  arrangement  of  the  valve  ham- 
mer, is  well  illustrated  by  that  on  the  water  engine  at  Bleiberg,  in 
Karinthia,  and  which  is  fully  described  in  G-erstners  "  Mechanics." 
Fig.  305  shows  this  arrangement  in  plan  and  elevation.  A  and  Al 
are  the  rods  of  the  working  pistons,  £DBl  is  a  balance  beam  con- 
nected by  chains  and  counter-chains  with  the  rods.  The  valve  ham- 
mer 6r,  and  its  wheel  FFV  on  the  horizontal  axis  M,  is  connected 
with  the  balance  beam  by  another  set  of  chains  FK  and  F^KV  An 
attentive  consideration  of  the  figure  shows  that  the  reciprocating 
motion  of  the  piston  rods  raises  the  hammer,  and  lets  it  fall  without 
hindrance  from  the  balance  beam  or  chains.  On  the  fall  of  the 
valve-hammer  wheel,  there  are  two  catches,  a  and  av  which,  when 
the  hammer  falls,  catch  upon  a  projection  on  the  horizontal  rod  LLV 
This  rod  LL  has  two  nobs  c  cv  into  which  the  handles  or  keys  of  the 
cocks,  K  and  Kl  are  set,  so  that  the  cocks  turn  through  an  angle  of 
90°,  when  the  hammer  in  its  fall  forces  the  bolt  6,  by  means  of  the 
catches  a  and  av  to  the  right  or  left.  This  method  of  moving  the 
cocks  is  necessarily  sudden,  and  gives  rise  to  violent  shocks,  so 
that  it  is  only  applicable  to  small  machines,  or  those  having  moderate 
falls. 

The  cocks  have  a  passage,  or  are  bored  through  the  axis,  and 
through  the  side.  Through  the  former  the  pressure  water  enters 
by  knee  pieces  0  and  Ol  into  the  barrel-shaped  bottom  pieces  N 


AUXILIARY  WATER-ENGINE  VALVE  GEAR.  3H 

and  N,  at  the  bottom  of  the  working  cylinders ;  and  through  the 
side  passage,  the  pressure  water  is  brought  to  the  cock  from  the 
cylinder.  In  order  that  only  as  much  water  may  be  used  as  is 
necessary  to  fill  the  space  passed  through  by  the  working  piston,  the 
discharge  is  made  to  take  place  under  water  into  special  reservoirs 
IF  and  Wr 

Remark.  The  engine  now  described  has  a  fall  of  260  feet,  stroke  6J  feet,  cylinder  7 
inches  diameter,  8  strokes  per  minute.  It  is  in  many  respects  an  imperfect  engine-  but 
it  is  economically  adapted  to  its  position.  We  have  not  only  to  consider  mechanical 
perfection  in  the  construction  of  engines  in  general,  but  we  have  to  weigh  well  the  cir- 
cumstances in  which  the  engine  is  to  work,  the  facilities  for  repair  in  the  particular 
locality,  and  the  relative  supply  and  demand  for  the  water  power. 

§  188.  Auxiliary  Water-Engine  Valve  Gear.— No  application  of 
spring-valve  gear  has  been  made ;  but  the  method  of  using  an 
auxiliary  water  engine  is  now  come  into  very  general  use.  The 
general  arrangement  of  such  an  auxiliary  engine  gear  is  shown  in 
Fig.  306,  as  applied  to  the  great  water-pressure  engine  in  the  Leo- 


Fig.  306. 


pold  shaft,  near  Chemnitz.  This  engine  has  two  cylinders,  C  and 
(7: ;  E  is  the  pressure  pipe,  A  the  discharge  pipe,  H  the  main  cock, 
K  a  quadrant  key  fastened  on  the  cock.  The  auxiliary  engine  has 
a  horizontal  cylinder  a  av  with  a  piston  b  on  the  piston  rod  c  cr 
The  piston  rod  is  connected  with  the  valve  rod  d  dl  by  cross  pieces, 
so  that  the  two  united  form  a  rectangular  frame.  The  valve  rod  is 
connected  with  the  quadrant  by  two  chains,  so  that  the  reciprocating 
motion  of  the  piston  b  communicates  a  rotary  motion  of  90°  to  the 
cock.  The  auxiliary  engine  is  worked  by  means  of  the  cock  h  Til 
lying  horizontally,  with  two  bores,  or  passages,  as  in  the  case  of  the 
main  cock  H.  The  little  pipe  e  communicating  with  the  pressure 
pipe  E,  takes  the  pressure  water  to  the  cock  h  hv  from  which  it 
passes  through  the  pipes  //,  to  one  side  or  the  other  of  the  piston 
b,  so  that  it  is  moved  backwards  and  forwards,  the  water  used  in 
each  alternate  stroke  being  discharged  by  the  other  passage  in  the 


312  AUXILIARY  WATER-ENGINE  VALVE  GEAR. 

cock,  and  thence  by  a  pipe  from  h.  The  small  cock  h  hl  is  turned 
by  the  double-handled  key  g  g^  connected  by  slender  chains  to  a 
double-armed  lever  parallel  to  it,  and  which  is  on  the  same  axis  as 
the  balance  beam  to  which  the  piston  rods  of  the  two  cylinders  are 

Fig.  307. 


attached.  The  whole  play^of  the  valve  gear  is  now  evident.  While 
the  working  piston  rises  and  the  other  descends,  the  cock  h  7^  is 
turned  by  the  lever  or  key  g  gv  thus  the  communication  between 
the  water  and  the  cylinder  a  at  is  opened  or  shut,  and  thus  power 
is  obtained  for  bringing  the  piston  i,  and  the  cock  H  into  the 
opposite  position,  so  that  the  first  working  cylinder  is  now  shut 
off  from  the  pressure  pipe,  and  the  second  put  in  communication 
with  it. 

Remark.  The  engine  in  the  Leopold  shaft  has  710  feet  fall  (Austrian  measure),  8  feet 
stroke,  1 1  inch  diameter  of  cylinder ;  each  piston  makes  3  strokes  per  minute. 

§  189.  The  working  of  the  valves  (Fig.  308)  by  means  of  an 
auxiliary  engine,  is  well  illustrated  by  that  of  the  double-acting, 
water-pressure  engine  at  Ebensee,  in  Salzburg ;  the  auxiliary  engine 
being,  in  this  case,  an  exact  model  of  the  working  engine.  CCl  is 
the  cylinder  of  the  principal  engine,  and  ecl  that  of  the  auxiliary. 
K  is  the  piston  of  the  one,  and  k  that  of  the  other  cylinder.  S  and 
Sl  are  the  valve  pistons  of  the  working,  and  *  and  sl  those  of  the 


THE  VALVE  CYLINDER. 


auxiliary 


313 

and 


uxiliary  engine.     EE,  (Fig.  308)  is  the  main  pressure  pipe,  a 
«,  the  pipe  communicating  with  the  auxiliary  engine.     Lastly,  . 
and  4,  are  the  orifices  of  discharge  of  the  main,  and  a  a.  those  of 
the  auxiliary  engine.     Thus  the  one  engine  is  an  exact  counterpart 


Fig.  308. 


of  the  other,  the  dimensions  being,  however,  very  different  in  the 
two.  The  valve  gear  of  the  auxiliary  engine  consists  in  the  canti- 
lever BD  attached  to  the  main  piston  rod  at  D — of  the  valve  piston 
rod  g  s,  connected  by  the  link/^r  to  the  rod  I  lv  on  which  there  are 
two  studs  placed,  so  that  the  lever  DB  catches  upon  them  a  little 
before  the  end  of  the  up  and  down  strokes,  respectively,  of  the  main 
piston,  and  thus  the  valve  piston  is  moved.  It  is  easy  to  trace  how 
this  motion  admits  the  pressure  water  alternately  above  and  below 
the  piston  JT,  so  as  to  raise  or  depress  the  valve  pistons  k  8l  8, 
giving  the  required  alternation  of  admission  of  the  pressure  water 
above  and  below  the  main  piston  K. 

Remark.  The  engine  at  Elxsnsee  has  a  fall  of  only  36  feet,  a  stroke  of  17  inches,  and 
a  cylinder  of  9£  inches  diameter.  It  makes  6  strokes  per  minute,  and  moves  two  dou- 
ble acting  pumps. 

§  190.   The  Valve   Cylinder.— In  the  larger  engines  of  recent 

date,  the  valve  pistons  of  the  main  cylinder  are  inclosed  in  the  same 

pipe,  or  cylinder,  as  the  piston  of  the  auxiliary  engine ;  and  in  some 

engines  the  counter-pressure  valve,  or  piston  balancing  the  pressure 

VOL.  n. — 27 


314 


THE  VALVE  CYLINDER. 


on  the  valve,  is  the  working  piston  of  the  auxiliary  engine,  and  thus 
great  simplicity  of  construction  is  attained. 

Fig.  309  shows  a  simple  arrangement  adopted  in  two  engines  in 
the  Freiberg  mining  district.  8  is  the  main  piston  valve,  and  G 
the  counter-pressure  piston,  O  an  intermediate  pipe  communicating 
with  the  main  cylinder,  E  the  entrance  for  the  pressure  water,  and 
A  the  orifice  of  discharge  for  the  water  used,  e  is  the  communica- 
tion with  the  valve  of  the  auxiliary  engine,  which  in  this  case  is  a 
cock.  The  piston  G  is  larger  than  S,  and,  therefore,  the  valve 


Fig.  309. 


Fig.  310. 


apparatus  S  G-  descends,  when  the  pressure  is  admitted  from  above 
sit  e,  and  ascends  when  the  pressure  water  is  cut  off  at  e,  and  the 
pressure  acts  underneath.  For  each  stroke  there  is  a  consump- 
tion of  a  certain  quantity  of  water  for  the  valves,  which  is  lost  for 
useful  effect.  This  amounts  to  the  contents  of  the  space  passed 
through  in  the  up  or  down  stroke.  In  the  construction,  now  under 
consideration,  this  is  not  so  little  as  in  some  others,  for  the  piston 
G  must  have,  at  least,  one  and  a  half  times  the  area  of  the  piston 
/$',  the  sectional  area  of  which  is  the  same,  or  even  greater  than  that 
of  the  pressure  pipe. 


SAXON  WATER-PRESSURE  ENGINE:  315 

The  system  of  valves  shown  in  Fig.  310,  is  that  of  the  Clausthal 
engine,  and  here  the  waste  of  water  is  less  than  in  the  last  men- 
tioned system.  For  there  are  three  pistons;  namely:  the  main 
valve  piston  AK,  the  counter  piston  <7,  and  the  auxiliary  piston  H; 
the  latter  being  somewhat  less  in  area  than  the  former.  The  water 
is  brought  into  the  valve  cylinder  by  the  pipe  e,  and  the  reverse 
motion  of  the  piston  is  effected  by  a  small  cock  through  which  the 
water  enters  before  coming  into  e,  and  through  which,  also,  when 
the  revolution  is  completed,  it  is  let  off.  The  cock  is  moved  by  link 
work,  by  means  of  a  tap  on  the  main  piston  rod. 

Remark.  The  engines  at  Clausthal  have  612  feet  fall,  diameter  of  cylinder  16$  inches, 
stroke  6  feet,  and  make  4  strokes  per  minute. 

§  191.  Saxon  Water-Pressure  Engine. — The  arrangement  and 
motions  of  a  double  cylinder  water-pressure  engine  may  be  clearly 
understood  by  a  study  of  a  sectional  view  of  the  engine,  erected  in 
the  Alte  Mordgrube,  near  Freyberg,  in  Saxony,  delineated  in  Fig. 
311.  CK  and  ClKl  are  the  two  working  cylinders,  Tif  and  Kl  being 
the  workipg  pistons,  S  and  T  are  the  two  valve  pistons,  W  is  the 
auxiliary  piston,  and  Sl  Tl  and  Wl  are  the  points  in  the  valve 
cylinder  ATW^  at  which  the  pistons  are  for  the  return  stroke  of 
the  working  pistons.  E  is  the  entrance  of  the  pressure  pipe  E^ 
into  the  valve  cylinder,  OS  is  the  intermediate  pipe  communicating 
with  the  one,  and  C^  the  pipe  communicating  with  the  other  work- 
ing cylinder.  A  is  the  orifice  of  discharge  of  the  one,  and  Av  that 
of  the  other  (this  latter  orifice  is  nearly  covered  by  the  piston  rod 
in  the  drawing).  The  two  piston  rods  BK  and  B^K^  are  connected 
by  a  balance  beam  (not  shown  in  the  figure),  so  that  as  the  one  piston 
ascends  the  other  descends.  It  is  hence  easy  to  perceive,  that,  for 
the  lower  position  of  the  valve  piston,  here  represented,  the  pressure 
water  takes  the  course  ESfi,  driving  the  piston  K  upwards,  and 
that  the  piston  Kl  is  pushed  downwards,  the  used  water  taking  the 
course  C1T1A1  to  the  discharge  orifice  Av. 

The  auxiliary  valve  consists  of  a  four-way  cock  A  (already  de- 
scribed) shown  at  I.  in  the  second  position,  and  external  elevation 
at  II.  This  cock  gives  passage  between  the  pipe  e  el  and  the  pres- 
sure pipe,  and  between  g  A  and  the  valve  cylinder. 

It  is  evident  that  in  the  one  position  of  A,  the  pressure  water 
takes  the  course  Eel  ehg  TF,  and  presses  down  the  auxiliary  piston 
W,  whilst  for  the  second  position  of  A,  the  pressure  water  is  shut 
off  from  W,  and  hence  the  ascent  of  the  valve  piston  system  STW, 
the  return  of  the  valve  water  through  g  A,  and  its  discharge  at  a  a,, 
can  take  place.  That  the  valve  piston  system  may  rise  w hen ^ the 
water  is  shut  off  from  IF,  and  may  descend  when  it  is  let  on,  it  is 
necessary  that  the  piston  T,  pressed  upwards  by  the  pressure  water, 
should  have  a  greater  sectional  area  than  the  piston  8,  which  is 
pressed  downwards  by  the  pressure  water;  and  also,  the  auxiliary 
piston  must  have  sufficient  area  that  the  water  pressure,  or  W  and 
>V  together,  may  exceed  the  opposite  pressure  on  T. 


316 


SAXON  WATER-PRESSURE  ENGINE. 


Fig.  311. 


HUELGOAT  WATER-PRESSURE  ENGINE.  317 

The  valve  gear  of  this  machine  is  composed  of  a  ratchet  wheel  r, 
a  catch  r  k,  a  rod  k  A,  and  a  bent  lever  h  cf  with  its  friction  wheel 
/,  and  the  two  wedge-formed  pieces  m  ml  set  on  each  piston  rod. 
The  catch  r  k  is  connected  with  the  axis  of  the  cock,  and  is  held 
by  a  small  balance  weight  q  in  its  place  on  the  ratchet  wheel. 
When  the  piston  K  has  reached  nearly  the  end  of  its  stroke,  the 
wedge  m  (or  m}]  passes  under  the  friction  wheel,  and  turns  the  lever 
fc  h  to  a  certain  extent,  so  that  the  rod  h  k  is  drawn  up,  and  the 
wheel  and  cock  h  are  turned  through  a  quadrant.  As  the  working 
piston  makes  its  return  stroke,  the  lever  falls  back  again,  and  the 
catch  slides  back  over  the  next  tooth  of  the  ratchet,  and  is  ready 
at  about  the  end  of  this  return  stroke  to  push  round  the  ratchet,  &c. 

Remark.  The  water-pressure  engine  in  the  Alte  Mordgrube,  has  a  fall  of  356  feet, 
a  stroke  of  8  feet,  18  inches  diameter  of  cylinder,  and  makes  4  double  strokes  per 
minute. 

§  192.  Iluelgoat  Water-pressure  Engine.  —  One  of  the  largest 
and  most  perfect  water-pressure  engines  hitherto  erected,  is  that  at 
Huelgoat,  in  Brittany.  It  is  a  single-cylinder,  single-acting  engine. 
Fig.  312  represents  the  essential  parts  of  this'engine,  and  its  valve 
gear.  00^  is  the  working  cylinder,  KKl  the  working  piston,  and 
BBl  the  piston  rod  working  through  a  stuffing  box  at  B.  In  the 
Saxon  engine,  the  piston  is  packed  by  a  single  sheet  of  leather;  but 
in  this  engine,  the  rim  of  the  piston  is  packed,  and  there  is  also  a 
sheet  of  leather,  held  in  its  place  by  a  ring.  The  valve  cylinder 
ASGr  is  united  to  the  working  cylinder  by  the  pipe  DDV  into  which 
the  pressure  pipe  opens  at  JE,  and  the  discharge  pipe  at  A.  To  the 
valve  piston  $,  a  counter-balance  piston  T  of  greater  diameter  is 
connected  by  the  rod  ST.  This  system  will,  therefore,  be  forced 
upwards  by  the  pressure  water,  if  a  third  force  be  not  brought  into 
play.  This  third  force  is,  however,  produced  by  bringing  the  pres- 
sure water  above  T,  through  the  pipe  el  ef,  and  in  order  to  use  only 
a  small  quantity  of  water  for  working  this  valve  system,  a  hollow 
cylinder  GH  is  placed  on  T,  passing  through  a  stuffing  box  at  ff, 
and,  therefore,  exposing  only  an  annular  area  to  the  pressure  of  the 
water. 

The  alternate  admission  and  exclusion  of  the  pressure  water  of 
the  hollow  space  g  g,  is  effected  by  an  auxiliary  valve  system,  re- 
sembling the  main  valve  system  in  every  respect ;  consisting  like  it 
of  a  valve  piston  *,  a  counter-balance  piston  t,  which  is  a  solid  piston 
passing  through  a  stuffing  box  at  h.  For  the  position  8  t  h,  shown 
in  our  figure,  the  pressure  water  has  free  circulation  through  c/to 
rj  ;  but  if  8  t  h  be  raised,  so  as  to  bring  «  above  /,  this  passage  is 
stopped,  and  the  valve  water,  in  the  hollow  space  gg,  escapes  through 
a  a,,  when  ST  goes  up.  Lastly,  to  derive  the  motion  of  the  auxiliary 
valve-piston  system  from  the  engine  itself,  there  is  let  into  the  work- 
ing piston  KKl  an  upright  rod  with  a  feather  edge  attached  to  the 
side.  This  feather  has  a  series  of  holes  drilled  in  it,  into  which 
catches  can  be  put  as  Xv  Xv  at  the  required  distance  apart.  The 
link  b  h  is  connected  to  two  levers,  centred  at  c  and  o,  and  connected 

27* 


318 


HUELGOAT  WATER-PRESSURE  ENGINE. 


Fig.  312. 


DARLINGTON  S  WATER-PRESSURE  EXtJINE. 


319 


by  the  link  I.     The  end  of  the  Fig.  313. 

one  lever  has  an  arc,  on  which 
there  are  two  projections  or 
catches  Yv  Yy  As  the  up 
stroke  of  the  piston  comes  to 
an  end,  the  catch  Xl  strikes 
on  Yv  and  thus  s  t  h  is  moved 
to  its  upper  position,  and  at 
nearly  the  end  of  the  down 
stroke  of  the  piston,  the  catch 
X2  strikes  Yy,  and  the  valve 
system  s  t  h  descends  to  a 
lower  position.  It  is  now  easy 
to  perceive  how  the  alternate 
positions  of  ST,  necessary  for 
the  reciprocating  motion  of 
the  piston,  KK^  are  pro- 
duced. 

§  193.  The  following  are 
the  details  of  the  construction 
of  Mr.  Darlington's  water- 
pressure  engine.  The  first 
engine  erected  in  England 
with  cylinder  or  piston  valves, 
was  that  put  up  in  the  Alport 
mines,  Derbyshire,  in  the  year 
1842.  This  was  a  single  cy- 
linder engine.  Its  success  was 
complete,  and  others  were 
erected  on  the  same  plan. 
But  in  1845,  a  combined  cylin- 
der engine  was  designed,  and 
erected  by  the  same  engineer, 
which  is  found  practically  to 
have  several  advantages  for 
such  large  supplies  of  water  as 
that  consumed  by  the  pump- 
ing engine,  of  which  are  sub- 
joined accurate  reductions  of 
the  working  drawings. 

Fig.  313  is  a  front  elevation 
of  the  combined  cylinder  en- 
gine. Fig.  314  is  a  sectional 
view,  and  Fig.  315  is  a  general 
plan.  PC,  is  the  bottom  of 
the  pressure  column,  130  feet 
high,  and  24  inches  internal 
diameter,  CO  are  the  combined  cylinders,  each  24  inches  diameter, 
open  at  top,  with  hemp-packed  pistons  a  (Fig.  314),  and  piston  rods 


320  DARLINGTON'S  WATER-PRESSURE  ENGINE. 

Fig.  3 14. 


DARLINGTON'S  WATER-PRESSURE  ENGINE. 


321 


m,  combined  by  a  cross  head  M,  working  between  guides  in  a  strong 
frame.  Ihe  admission  throttle  valve  is  a  sluice  valve,  shown  at  o 
±ig.  616,  and  between  the  letters  b  and  e  and  Fig.  315.  The  main 


or  working  valve,  is  a  piston  g,  18  inches  in  diameter,  Fig.  314, 
with  its  counter  or  equilibrium  piston  above.  The  orifice  for  the 
admission  of  the  pressure  water  is  between  the  two  pistons.  The 
intermediate  pipe  a  is  a  flat  pipe,  into  which  numerous  apertures 
lead  from  the  valve  cylinder  (seen  immediately  under  g,  Fig.  314). 
The  valve  piston  is  in  the  position  for  discharging  the  water  from 
the  cylinders  through  the  pipe  6,  Fig.  314,  by  the  sluice  valve  k. 

The  valve  gear  is  worked  by  an  auxiliary  engine  A,  by  means  of 
the  lever  v.  The  auxiliary  engine  valves,  are  piston  valves  in  the 
valve  cylinder  «,  Figs.  314  and  315,  communicating  with  the  pres- 
sure pipes  by  a  small  pipe,  provided  with  cocks,  as  shown  in  Fig. 
315.  The  motion  of  the  auxiliary  engine  valves  is  effected  by  a  pair 
of  tappets  £',  t",  set  on  a  vertical  rod  attached  to  the  cross  head  n. 
These  tappets  move  the  fall  bob  b,  by  means  of  the  canti-lever  «, 
Fig.  313,  the  other  end  of  the  lever  being  linked  to  the  rod  *,  Fig. 
314,  which  again  is  linked  to  the  auxiliary  piston  valve  rod. 

The  play  of  the  machine  is  now  manifest.  It  is  in  every  respect 
analogous  to  the  Harz  and  Huelgoat  engines,  described  above. 
The  average  speed  of  the  engine  is  140  feet  per  minute,  or  7  double 
strokes  per  minute.  This  requires  a  velocity  of  something  less  than 
2|  feet  per  second  of  the  water  in  the  pressure  pipes ;  and  as  all  the 
valve  apertures  are  large,  the  hydraulic  resistances  must  be  very 
small.  The  engine  is  direct  acting,  drawing  water  from  a  depth  of 
135  feet,  by  means  of  the  spear  w,  w.  Figs.  313  and  314.  The 
"box,"  or  bucket  of  the  pump,  is  28  inches  in  diameter,  so  that 
the  discharge  is  266  gallons  per  stroke,  or,  when  working  full  speed, 


322  BALANCE — THROTTLE  VALVES. 

1862  gallons  per  minute.  The  mechanical  effect  due  to  the  fall 
and  quantity  of  water  consumed  is  nearly  140  horse  power.  The 
mechanical  effect  involved  in  the  discharge  of  the  last-named  quantity 
of  water  is  nearly  74  horse  power,  so  that,  supposing  the  efficiency  of 
the  engine  and  pumps  to  be  on  a  par  with  each  other,  the  efficiency 
of  the  two  being  (§  203),  ^  =  71,15,  the  efficiency  of  the  engine 

alone  >j  =  —  ~—  = — - —  =  ,85,  o'r,  in  the  language  of   Cornish 

engineers,  85  per  cent,  is  the  duty  of  the  engine. 

The  cost  of  maintenance,  grease,  &c.,  of  the  engine,  is  only  ,£40 
per  annum.  In  every  particular,  it  redounds  to  the  credit  of  Mr. 
Darlington's  skill  as  an  hydraulic  engineer. 

Balance. — For  regulating  the  motion  of  water-pressure  engines, 
several  auxiliary  arrangements  are  necessary,  which  we  shall  explain 
hereafter.  The  ascent  and  descent  of  the  working  piston  is  regu- 
lated by  an  arrangement  called  a  balance,  or  counter-balance,  which 
aids  the  motion  of  the  piston  in  the  one  direction,  and  retards  it  in 
the  other,  so  that  the  working  of  the  machine  goes  on  with  a  nearly 
uniform  velocity.  In  the  double-cylinder  engine,  the  balance  is 
effected  by  a  simple  beam,  connecting  the  two  cylinders.  In  the 
double-acting,  single-cylinder  engine,  a  fly  wheel  is  necessary,  and 
in  the  single-acting,  single-cylinder  engine,  a  counterbalance  Aveight, 
either  of  a  solid  body,  or  of  a  column  of  water,  an  hydraulic  balance, 
is  employed.  On  the  subject  of  "  Regulators  of  Motion,"  we,  for 
the  present,  make  only  a  few  general  remarks.  The  mechanical 
balance  consists  of  a  beam  with  a  weight  at  one  end,  and  having  the 
other  end  attached  to  the  piston  of  the  engine,  so  that  the  weight 
assists  during  the  working  stroke  of  the  piston  to  counterbalance 
the  piston  and  rods;  and  during  the  down  stroke,  or  discharge  of  the 
used  water,  prevents  the  too  rapid  return  of  the  piston  and  rods ; 
the  adjustment  being  such  as  to  allow  of  the  discharge  stroke  being 
made  in  about  half  the  time  that  the  working  stroke  occupies. 
The  hydraulic  balance  consists  of  a  second  column  of  pipes,  which 
ascends  from  the  discharge  orifice  to  such  a  height,  that  the  water  it 
contains  counterbalances  the  extra  weight  of  the  piston  and  rods. 
The  machine  at  Huelgoat,  and  also  those  at  Clausthal,  have  hy- 
draulic balances. 

There  is  evidently  neither  loss  nor  gain  of  effect  by  the  use  of  a 
counter-balance,  save  by  the  prejudicial  resistances  they  give  rise 
to  in  their  motion.  The  balance  beam  has  the  advantage  that  its 
balance  weight  can  be  varied  as  required ;  and  the  hydraulic  balance 
has  the  advantage  of  simplicity,  when  other  circumstances  do  not 
interfere  with  its  application. 

§  194.  Throttle  Valves. — The  cocks  or  throttle  valves  of  water- 
pressure  engines  are  important  organs,  their  function  being  to  regu- 
late the  supply  of  water  to,  and  its  discharge  from,  the  engine — that 
is  the  speed  of  the  engine.  These  valves  must  have  a  prejudicial 
effect  on  the  efficiency  of  the  engine,  and  yet,  they  are  a  necessary 
evil.  -  In  order  to  regulate  the  ascent  and  descent  of  the  working 


WATER-PRESSURE  ENGINES.  323 

piston  and  of  the  valve  pistons,  there  are  necessary,  four  cocks  or 
valves — one  in  the  pressure  pipe,  and  one  in  the  discharge  pipe  (as 
Z,  Fig.  312) ;  also  a  cock  in  the  pipe  leading  the  valve  water  above 
the  auxiliary  piston,  and  another  similar  in  the  pipe  which  discharges 
the  water  used  in  the  valves,  as  e  and  a  in  Fig.  312. 

To  get  the  highest  efficiency  from  a  water-pressure  engine,  its 
work  should  be  such  as  to  render  any  contraction  of  the  pressure 
pipes,  by  a  throttle  valve,  unnecessary  for  its  uniform  motion.  If, 
however,  the  useful  effect  of  the  engine  is  greater  than  is  required 
by  the  work  to  be  done,  the  excess  must  be  taken  away  by  checking 
the  supply  by  means  of  the  throttle  valve,  or  by  shortening  the 
stroke  of  the  engine. 

If  it  be  an  object  to  save  water,  the  latter  means  is  the  best  when 
possible,  because  the  efficiency  of  the  machine  is  not  thereby  inter- 
fered with. 

A  change  in  the  length  of  stroke  of  the  piston  is  easily  effected 
by  altering  the  position  of  the  catches  on  the  rod  Xv,  X3,  Fig.  312. 
The  nearer  X}  and  X2  are  brought  together,  the  earlier  the  revers- 
ing of  the  stroke  ensues  ;  and,  therefore,  the  shorter  is  the  stroke  of 
the  working  piston. 

§  195.  Mechanical  Effect  of  Water-pressure  Engines. — In  comput- 
ing the  effect  of  water-pressure  engines,  we  shall  make  use  of  the 
following  symbols : — 

F  =  the  area  of  the  working  piston. 

Fl  =  the  area  of  the  pressure  pipes. 

d  =  the  diameter  of  the  working  piston. 

dl  =  the  diameter  of  the  pressure  pipe. 

d2  =  the  diameter  of  the  discharge  pipe. 

h  =  the  fall  from   surface  of  reservoir  to  surface  of  water  in 
discharge  channel. 

A2  =  the  vertical  distance  from  the  surface  of  reservoir  to  the  sur- 
face of  piston  at  half  stroke. 

Jiz  =  the  distance  from  surface  of  discharged  water  to  the  piston 
at  half  stroke. 

«    =  the  stroke  of  the  piston. 

Zt  =  length  of  pressure  pipe. 

lz  =  length  of  discharge  pipe. 

v   =  mean  velocity  of  piston^ 

vl  =  mean  velocity  of  water  in  pressure  pipe. 

v2  =  mean  velocity  of  water  in  discharge  pipe. 
We  shall  assume  the  engine  to  be  single  acting,  making: 

n  —  the  number  of  strokes  per  minute. 

Q  =  the  quantity  of  water  used  per  second. 

The    mean    pressure    of  the   water   on  the   piston  surface  Jf  i 
PI  =  F  ;ti  y,  and,   therefore,  the  mechanical  effect  produced  per 
stroke,  prejudicial  resistances  neglected,  is  P,  *  =  Fsh  y,  and  per 
minute  n  Ps  =  nFs  h,  y,  and,  therefore,  the  mean  effect  per  se- 
cond, is: 


324  FRICTION  OF  THE  PISTON. 


In  the  return  of  the  piston,  the  mean  effective  resistance  is: 
P2  =  F  h2  y,  and,  therefore,  the  mechanical  effect  consumed  is : 
jP3  s  =  F  h2s  y,  and  hence  the  loss  of  effect  per  second  is : 
L2  =  Q  A2  y,  and,  therefore,  the  effect  available 

L  =  Lt  —  L2  =  Q  ( Aj  —  A2)  y  =  Q  A  y, 
as  in  many  other  hydraulic  recipient  machines. 

This  formula  is  evidently  not  changed  should  the  working  piston 
not  fill  up  the  cylinder,  i.  e.,  supposing  a  plunger  is  used,  round 
which  there  is  a  free  space,  or  supposing  the  piston  does  not  descend 
to  touch  the  bottom  of  the  cylinder.  Nor  would  the  circumstance 
of  the  discharge  taking  place  below  the  mean  position  of  the  piston 
— that  is,  of  Aa  being  negative,  or  A  =  Aj  +  As  alter  the  formula. 
F  is  the  area  of  a  section  of  the  piston  at  right  angles  to  its  axis, 

or  F  =  — —  ,  and,  therefore,  the  form  of  the  piston  can  have  no 

effect. 

§  196.  Friction  of  the  Piston.  —  Of  the  prejudicial  resistances, 
the  friction  of  the  piston  is  a  principal  one.  As  there  are  no  accu- 
rate experiments  on  this  subject,  we  must  content  ourselves  by  esti- 
mating it  from  the  pressure  of  the  water,  and  a  co-efficient  of  fric- 
tion ascertained  in  the  nearest  possible  analogous  circumstances.  If 
the  packing  be  on  the  hydrostatic  plan,  the  force  with  which  each 
element  e  of  the  packing  is  pressed  against  the  cylinder  during  the 
up  stroke  is  =  e  A,  y,  and  during  the  down  stroke  it  is  =/e  A2  y,  and 
hence  the  friction  =  f  e  A,  y,  and /eA2  y,  respectively.  The  total 
friction  will  be  the  sum  of  the  frictions  of  all  the  elements,  or  of  the 
area  of  the  whole  packing.  If  the  breadth  of  the  packing  be  b, 
then  A  d  b  is  the  area,  and  then  the  piston  friction  is  Rl=frtdb  h1y 
for  up  stroke,  and  Jt2  =/*  d  b  A3y  for  down  stroke. 

It  is  convenient  to  express  the  various  prejudicial  resistances  in 
terms  of  a  column  of  water  of  the  area  of  the  piston,  and  whose 
height  A3  or  A4  is  the  head  lost  (in  the  present  case)  by  the  friction 
of  the  piston.  Let  us,  therefore,  put: 


or  putting  —  for  F,  we  have  1*3  =  /  b  hv  and  ^A  =  fb  A2  ;  and 
hence  the  loss  of  fall,  corresponding  to  the  friction  of  the  piston 

A3  =  4/^A15  andA4=4/*  A, 

a  a 

If  we  deduct  these  heights,  we  get  for  the  mean  power  during  the 
up  stroke: 


and  during  the  down  stroke  : 


HYDRAULIC  PREJUDICIAL  RESISTANCES.  325 

and  hence  the  resultant  mean  effect  : 


If  the  rising  pipe  height  h2  =  0,  or  be  very  small,  then  we  have 
more  simply 


"We  see  from  this  that  the  loss  of  effect  from  friction  of  piston  is 

so  much  the  greater,  the  greater  —  -  and  _J  are,  that  is,  the  greater 

h          h 
the  head,  and  the  greater  the  counter-balance  head. 

To  reduce  this  friction,  the  packing  should  not  have  unnecessary 

width.     In  existing  machines  -  ==  0,1  to  0,2.     The  co-efficient  of 
friction  is  to  be  taken  as  determined  by  Morin,  /=  0,25.    This  being 

assumed,  we  see  that  4/-  =  0,1  to  0,2,  or  that  the  friction  of  the 

d 

piston  absorbs  from  10  to  20  per  cent,  of  the  whole  available  power. 
§  197.  Hydraulic  Prejudicial  Resistances.  —  Another  source  of 
loss  of  effect  in  water-pressure  engines,  is  the  friction  of  the  water 
in  the  pressure  and  discharge  pipes.  According  to  the  theory  given 
in  Vol.  I.  §  329,  the  pressure  height  or  head  corresponding  to  this 
loss,  ?  being  the  co-efficient  of  friction,  is 

h  =  £  .  —  .  —  .     This  applied  to  the  pressure  pipe,  becomes 
d     2g 

hs  =  ?  .  i  .  JL,  and  applied  to  the  discharge  pipe  it  is 
dl     2<jf 

A6=  f  .  i  .  JL.     But  the  quantity  of  water,  is 
«3     *9 

!*L  .  Vl  =  5A2  .  *  =  1*  v,  therefore, 
444 

d,2  v,  =  d*  v2  =  d2  v,  or  vt  =  (j\*  v,  and  v2  =  ^  v,  and,  hence, 

wemayputl     .  «,*  *•  '        .  'i£  + 
*-*'  *••*•?*?*&?'* 

and  for  velocities  (v.  and  vz)  of  from  5  to  10  feet, 

f  =  0,021  to  0,020. 

In  order  to  reduce  these  resistances,  the  pipes  must  be  of  as  great 
diameter  as  possible,  and  the  number  of  strokes  as  few  as  possible. 
The  motion  of  water  in  the  pipes  of  a  water-pressure  engine  is 
different  from  that  in  ordinary  conduit  pipes,  inasmuch  as  in  the 
former  the  velocity  continually  varies,  whilst  in  the  latter  it  is 
sibly  uniform. 
VOL.  II.—  28 


326  HYDRAULIC  PREJUDICIAL  RESISTANCES. 

Hence  the  inertia  of  the  water  plays  a  more  conspicuous  part  in 
the  one  than  in  the  other.  In  order  to  put  a  mass,  Jf,  into  motion 
with  a  velocity  v,  there  is  required  to  be  expended  an  amount  of 

mechanical  effect  represented  by ;  and  hence  to  communicate 


to  the  column  of  water  in  the  pressure  pipes  a  velocity  vv  the 
weight  being  Jf^  ^  y,  there  is  required  an  amount  of  mechanical 

effect=jPj?jy  .  -J-.     If  the  water  column  be  cut  off  from  the  working 

cylinder  only  at  the  end  of  the  stroke  of  the  piston,  this  amount  of 
effect  would  not  be  lost,  for  this  column  would  restore,  or  give  back 
the  mechanical  effect,  during  the  gradual  cessation  of  the  piston's 
motion;  but  the  cutting  off  of  the  water  pressure  from  the  working 
piston  takes  place,  although  near  the  end  of  the  stroke,  yet  gra- 
dually and  while  the  piston  is  in  motion,  so  that  the  working  piston 
and  column  of  water  come  to  rest  at  the  same  instant  ;  and  hence 
the  valve  piston  causes  a  gradual  absorption  of  all  the  vis  viva  of 
the  water  column  during  the  first  half  of  its  ascent,  inasmuch  as  it 
brings  a  gradually  increasing  resistance  in  the  way,  by  gradually 
decreasing  the  passage,  and  hence  we  may  assume  that  the  mechani- 

cal effect  due  to  inertia,  Fl  Z,  y  .  J-  is  lost  at  each  stroke. 

If  we  introduce  vl  =  —  v,  and  F1  =  ^-L,  then  we  have  for  the 

above  amount  of  mechanical  effect  -  .  —  i  y  .  —  ,  and,  hence,  the 

4        d*        2g 
mean  effort  during  the  whole  stroke  a, 

«d«      d"^  V* 

--4T'^8V'%> 

and  the  corresponding  loss  of  fall  or  pressure  head  : 

A       **i     * 
7=^'¥ 

A  loss,  that  would  be  expressed  in  a  similar  manner,  takes  place 
on  the  return  stroke,  during  which  the  water  is  forced  out  of  the 
cylinder  with  a  velocity  v2,  and  the  vis  viva  communicated  to  it  at 
the  commencement  of  the  stroke  is  of  course  lost  to  the  efficiency 
of  the  engine.  The  pressure  head  lost  would  be  : 


To  keep  these  two  losses  by  inertia  as  small  as  possible,  it  is  re- 
quisite to  have  the  pressure  and  discharge  pipes  of  greatest  diameter, 
and  least  length  possible,  to  have  a  small  velocity  of  the  piston,  and 
a  long  stroke. 

Remark.  To  mitigate  or  to  get  rid  of  the  prejudicial  effect  of  shock,  which  the  sudden 
cutting  off  of  the  water  gives  rise  to,  an  air  vessel  has  been  introduced  in  many  engines, 
at  the  lower  end  of  the  pressure  pipes,  and  near  the  valves.  This  is  a  cylinder  filled  with 
compressed  air,  analogous  to  the  air  vessels  on  fire  engines.  The  air  in  this  case  absorbs 
the  excess  of  vis  viva  in  the  water,  being  compressed  by  it;  and  the  air  expanding  again, 


SECTIONAL  AREAS.  327 

restores  this  vis  viva  at  the  commencement  of  the  next  stroke,  the  water  being  forced 
from  the  air  vessel  into  the  working  cylinder,  nearly  as  if  under  the  original  hydrostatic 
pressure.  In  the  application  of  this  arrangement  to  machines  having  very  great  falls, 
the  air  in  the  vessel  has  been  found  to  mix  with  the  water,  so  that  it  is  gradually  removed 
from  it  entirely.  To  prevent  this,  either  a  piston  must  be  fitted  into  the  air  cylinder,  or 
air  must  be  continually  supplied  to  it  by  a  small  air  pump  to  make  up  the  absorption  of 
it  by  the  water. 

§  198.  Changes  in  direction  and  in  sectional  areas  of  the  various 
pipes  of  a  water-pressure  engine  are  further  causes  of  diminished 
efficiency.  Although  these  losses  may  be  calculated  by  the  formulas 
given  in  the  third  and  fourth  parts  of  the  sixth  section  of  the  first 
volume,  it  appears  necessary  that  we  should  here  bring  together  the 
formulas  to  be  applied. 

In  the  pressure  and  discharge  pipes,  there  are  bent  knee  pieces, 
the  motion  of  the  water  through  which  involves  a  loss  of  head,  which 
may  be  expressed,  according  to  Vol.  I.  §  334,  by  the  formula 

h  =  £  .  —  .  —  .     Here  0  is  the  arc  of  curvature,  generally  =  ^,  ^  is 

*     % 

a  co-efficient  depending  on  the  ratio  between  the  radius  r  of  the  sec- 
tional area  of  the  pipe,  and  the  radius  of  curvature  of  the  axis  of  the 
pipe,  and  which  may  be  calculated  by  the  formula 

^  =  0,131  +  1,847  (-)5>  or  may  be  taken  from  the  tables  given  at 

the  place  cited.  For  a  bend  in  the  pressure  pipe,  the  head  due  to 
the  resistance  is 


and  for  a  bend  in  the  discharge  pipe  : 

h    _C      P*     ^2_,f2     /A4    ^ 

>  *«  "£.',JJT;*  -   w  'ty 

At  the  entrance  of  the  water  into  the  valve  cylinder,  as  well  as  at 
its  discharge  from  it,  the  water  is  suddenly  turned  aside  at  a  right 
angle,  exactly  as  in  an  elbow,  or  rectangular  knee  piece.  There  is, 
therefore,  a  loss  of  head  in  this  case,  which,  according  to  Vol.  I.  § 

333,  may  be  put:  h  =  0,984  —  ,  or  almost  equal  to  ^-  .     For    uni- 
formity's sake,  we  shall  put  this  loss  of  head  for  the  pressure  pipe  : 


and  for  the  discharge  pipe  : 


Sudden  changes  in  sectional  area,  as,  for  example,  at  the  entrance 
and  discharge  of  the  water  into  and  from  the  working  cylinder, 
give  rise,  in  like  manner,  to  a  loss  of  pressure  head.  According 
to  Vol.  I.  §  337,  such  a  loss  is  determined  by  the  formula, 

Ji=  (—  —  iV—  •     For  the  entrance  of  the  water  into  the  work- 
\F,        )  fy 


328  FORMULA  FOR  THE  USEFUL  EFFECT. 

ing  cylinder,  this  formula  applies  directly,  if  F  and  F1  be  the  areas 
of  the  cylinder  and  pressure  pipes  respectively.     For  the  discharge 

~UI  -t 

on  the  other  hand  —  =  -,  in  which  a  is  the  co-efficient  of  contrac- 

Fl         a 

tion.     If  a  =  0,6,  then  (  --  1\  =  |;  and  hence  the  head  due  to 

the  resistance  to  the  entrance  of  the  water  into  the  cylinder  : 
,         /F       ,\* 

h= 


and  for  the  discharge  : 

•-e 

For  simplicity's  sake,  however,  we  shall  put 


7i  d? 


so  that,  when  F=  _  is  introduced,  and^  =      "*  ,  then 


To  avoid  loss  of  effect  by  sudden  variations  of  velocity  generally, 
the  intermediate  pipes,  and  parts  of  the  valve  cylinder  through 
which  the  water  passes,  should  have  the  same  area  as  the  pressure 
and  discharge  pipes,  or,  at  all  events,  the  intermediate  passages 
should  gradually  widen  out  to  the  area  of  the  main  pipes. 

There  are  further  special  losses  of  effect  occasioned  by  the  cocks 
and  throttle  valves.  These  are  to  be  calculated  by  the  formula 

A  =  f  5  .  __  and  the  co-efficients  ?5  depend  on  the  position  or  angle 

of  the  cocks,  &c.,  and  are  to  be  taken  from  the  tables,  Vol.  I.  §  340. 
Hence  for  the  ascent  of  the  working  piston  : 

*,.  -  S*  -         4  -        and  for  the  descent,  h16  =  £ 


By  setting  the  regulating  cock  or  valve,  the  co-efficient  of  resist- 
ance may  be  varied  to  any  amount  from  0  to  oo  ,  or  any  excess  of 
power  may  be  absorbed,  and  the  velocity  of  the  piston  checked  at 
pleasure. 

§  199.  Formula  for  the  Useful  Effect.—  If  in  the  mean  time  we 
leave  the  valves  out  of  consideration,  we  can  now  construct  a  for- 
mula representing  the  useful  effect  of  a  water-pressure  engine.  The 
mean  effort  during  an  ascent  of  the  piston  is 

P  =  [A,  -  (h?  +  hs  +  A7  +  hg  +  hn  +  A13  +  AJ]  Fv, 
and  the  resistance  in  the  descent  : 

P1  =  (A2  +  h4  +  h6  +  h8  -f  A10  +  Au  +  A14  +  A18)  FT, 
and  hence  the  effect  for  a  double  stroke: 


FORMULA  FOR  THE  USEFUL  EFFECT.  329 

and  the  mechanical  effect  produced  per  second  : 

L  =  [h,—  (h2  +  h3  +  \  +  .  .  .  +  A16)]  .  2L  .  F8y. 
If,  again,  we  put: 


then  we  may  express  the  useful  effect  very  simply  and  comprehen- 
sively, by 


On  account  of  the  greater  length  of  the  pressure  pipes,  x1  is  con- 
siderably more  than  *2;  and,  therefore,  the  time  for  the  up  stroke 
Jj  is  usually  allowed  to  be  longer  than  that  for  the  down  stroke  ty 

If  we  make  the  ratio  -i  =  v  =  f  ,  then 

•*•  »       260"       ,  .          1        60" 

t,  =  _—  --  ,  and  tz=  ——  .  —  ; 

v  +  1       n  v  +  1      n 

and  if  we  retain  v  as  the  value  of  the  mean  velocity  of  a  double  stroke 

-  £_  —  *>ns   then  the  mean  velocity  during  the  up  stroke 
<!  +  t2       60" 

_«_*•+!     ns  ___  v  +  1     w 

=  Fj  =  ~T~  '  60  ""  ~~1T  '  2' 
and  that  during  the  down  stroke 


and,  hence,  the  useful  effect  may  be  expressed  more  generally  : 


or,  introducing    L  .  Fs=  Q, 


Q     4  Q 

or,  introducing  v  =s  -^  =  — -y, 


330  FORMULA  FOR  THE  USEFUL  EFFECT. 


In  the  double-acting,  water-pressure  engine,  the  mechanical  effect 
produced  is  of  course  doubled.  • 

This  formula  shows,  very  clearly,  that  the  useful  effect  of  a  water- 
pressure  engine  is  greater,  the  greater  dj  d^  and  d2  are,  or  the  wider 
the  cylinder  and  pipes.  It  is  also  demonstrable,  by  aid  of  the 
higher  calculus,  that  for  a  given  number  of  strokes  the  useful  effect 
is  a  maximum,  or  the  prejudicial  resistances  are  a  minimum,  when 

-^-  =  *     that  is,  when  *  =  3  pJ3r.     If,   for   example,  d,  =  d,, 

'     «1  «2  \  *2  «i 

and  x1  =  8  *2,  then  v  =  4/8  =  2,  or  the  time  for  the  up  stroke  would 
be  double  that  for  the  down  stroke.  By  applying  a  balance  beam, 
attached  to  the  working  piston  rod,  this  ratio  v,  between  the  time  for 
the  up  and  down  stroke,  may  be  adjusted  by  the  counter-balance 
weight  applied.  Any  regulation  by  means  of  the  throttle  valve,  or 
cocks,  on  the  pressure  or  discharge  pipes,  can  only  be  effected  at 
the  cost  of  useful  effect,  as  by  these  a  loss  of  power  measured  by  f& 
is  occasioned,  and  which  increases  in  proportion  as  the  passages  are 
contracted. 

If  the  mechanical  effect  required  be  less  than  the  best  effect  of 
the  engine,  the  excess  must  be  destroyed  or  checked  by  the  throttle 
valves. 

Example.  It  is  required  to  make  the  calculations  necessary  for  establishing  a  single- 
acting,  single  cylinder,  water-pressure  engine  for  a  fall  h  =  350  feet,  and  a  quantity  of 
water  Q  ^  1  cubic  foot  per  second. 

Suppose  v  the  mean  velocity  of  the  up  and  down  stroke  ^  1  foot,  then  for  its  area, 

we  have  F  =  —  =  — I —  =  2  square  feet ;  and  if  we  arrange  that  the  water  shall 

v  1 

move  through  the  pressure  and  discharge  pipes  with  a  velocity  »,  =  ?,,=  5  feet,  then 

for  the  section  of  these  pipes,  we  have  jF,  =  — =  |  =  0,4  square  feet  Hence  the 
diameter  of  the  working  piston,  d=  /i^=  p.  =  1,5958  feet;  and  that  of  the  pres- 
sure and  discharge  pipes,  rf,  =  d,=  / L=  /— =  0,71364  feet.  For  simplicity 

and  certainty,  we  shall  assume  d  =  20  inches,  and  dl  =  d2  =  7  inches. 

If,  for  counter-balancing  the  rods,  &c..  we  carry  up  .the  discharge  pipe  50  feet  above 
the  mean  height  of  the  piston,  or  make  h2=  50  feet,  then  A,  =  A-f-  h^  =  400  feet.  We 
shall  assume  further,  that  the  total  length  of  pressure  pipe  Z,  =  450  feet,  and  that  of  the 
discharge  pipe  1^=66  feet.  For  a  diameter  of  20  inches, 

F  =  —  =  -  .  —  =  2,182  square  feet  ...t>  =  _  = =0,9166  feet 

Suppose  we  have  4  strokes  per  minute,  then  the  length  of  stroke 
.  _  60  v  _  60  .'  0,9166  _  fi  S7J!S  f<M>t 

2  n  "8 

If,  again,  we  suppose  the  width  of  the  packing  of  the  piston  b  =  $d  =  2£  inches,  we  get 
as  the  pressure  height  absorbed  by  the  friction  of  the  piston : 

4  .  0,25  .  i  (400  +  50)  =  -15°.  =  56,25  feet, 


ADJUSTMENT  OF  THE  VALVES.  331 

or  there  remains,  after  deducting  the  piston  friction,  the  head  350 56  25  =  293  75 

feet     To  calculate  the  hydraulic   resistances,  we  must,  in  the  first  place, 'determ ine  «, 
and  «2.     That  for  the  pressure  pipe, 

and  that  for  the  discharge  pipe : 


and  in  these  expressions  : 

£  =  0,021,  ^.=i^.  =  600,and^  =  ^=88; 
"i          *  d»       * 

therefore,  f  *L  =  0,021  .  600  =  12,6,  and  fk  =  0,021  .  88  =  1,85.     Again, 


/9Y      450 
\20/      6,87 


450  and  £A       /9_V  _66_ 

ft        W    6,87 


If  we  further  assume,  that  the  bends  in  the  pipes  have  radii  of  curvature  a  =  4r,  or 

if  —  =  i,  we  have  as  the  co-efficient  of  resistance  in  bends  : 
a 

C,  =  0,131  -f  1,847  fLjjl  =  0,145,  and  if  the  aggregate  angle  of  deflexion  by  curves 
in  the  pressure  and  discharge  pipes  =  270°,  or  if: 

S     =  °22 


If,  further,  the  water,  before  and  after  its  work  is  done  in  the  cylinder,  makes  two  rec- 
tangular deviations  in  its  progress  through  the  valve  cylinder,  we  have,  in  the  formulas 
for  *,  and  «,„  £2  =  2  .  1  =  2  ;  and  if  the  valve  cylinder  is  of  the  same  diameter  as  the 
pressure  and  connecting  pipes,  the  co-efficient  of  resistance  for  the  up  stroke 

*r  f3  =  f  1  —  frLV 1  "=  (1  —  0,2025)2  =  0,64,  whilst  for  the  down  stroke  £4  =  f  =  0,44. 

If  the  throttle  and  other  passage  valves  be  fully  open,  then  f  5  =  0,  and,  therefore,  we 
have  xt  =  12,60  -f  13,26  -f  0,22  +  2,00  -f  0,64  =  28,72,  and 

«,  =  1,85  +  1,94  +  0,22  -f  2,00  -f  0,44  =  6,45 
Lastly,  we  have  the  best  ratio  of  the  times  for  the  up  stroke  and  down  stroke  : 

,  =  3  fe  =    I28'72  =  1,646,  or  nearly  5  to  3. 

>J«.,       V    6,45 
By  introducing  these  values,  we  get  the  height  of  column  remaining: 


=  293,75-16,79.  0,0155  .  ^-1^  =  293,75-2,37  =  291,38  feet 
From  this  we  get  the  efficiency  of  this  engine,  neglecting  the  mechanical  effect  required 
for  working  the  valves,  «  =  ^^  =  0,832,  and  the  useful  effect  : 
L  =  291,38  .  1  .  62,5  =  18211  feet  Ibs.,  or  3,1  horse  power,  nearly. 

§  200.  Adjustment  of  the  Valves.—  The  arrangement  and  proper 
adjustment  of  the  valves  is  a  most  important  part  of  the  water-pres- 
sure engine.     As  in  all  the  engines  we  have  described  piston  valves 
are  used,  we  shall,  in  what  follows,  confine  ourselves  to  the  con 
tion  of  these  arrangements. 

We  shall  first  consider  the  system  having  two  pistons,  as  i 
some  of  the  Saxon  engines,  and  represented  in  Fig.  61b. 


332 


ADJUSTMENT  OF  THE  VALVES. 


If  we  assume  that  the  valve  piston  S  is  pressed  upwards  with  a 

mean  pressure  ht,  and  downwards  with  a  pressure  A2;  and  if  the 

height  of  the  counter  piston  Gr  above  S  =  e,  and, 

Fig.  316.  therefore,  the  height  of  the  hydrostatic  column  under 
Gr  =  Jiz  —  e,  and  that  above  Gr  according  as  the 
water  is  let  on  or  shut  off,  hl  —  e,  or  h2  —  e.  If 
further,  dl  =  the  diameter  of  S,  and  d2  =  that  of  Gr, 
and  we  shall  assume  that  the  packing  of  the  two  pis- 
tons consists  of  leather  discs  pressed  together,  and  that 
they  are  about  the  same  height  or  thickness.  If,  now, 
this  piston  valve  system  be  up,  as  shown  in  Fig.  316, 
the  letting  on  of  the  pressure  water  above  Gr  would 
occasion  a  descent  of  the  valves,  and,  therefore,  the 
difference  of  the  water  pressure  on  S  and  Gr,  in  com- 
bination with  the  weight  R  of  the  system,  must  be 
sufficient  to  overcome  the  friction  of  the  piston  S  and 
Gr.  The  pressure  downwards  on 


S|      and  the  counter  pressure  under 


The  downward  pressure  on  S 


h2  y,  and  the  counter  pressure 


under  S  = 
down: 


y,  and,  hence,  the  power  to  push  the  system 


or  the  fall  A,  —  h2  being  represented  by  h  : 


The  friction  of  the  pistons,  even  though  they  be  not  on  the  hydro- 
static principle,  is  proportional  to  the  circumference  of  the  piston, 
and  to  the  difference  of  pressure  on  the  two  sides  of  the  piston,  and 
may  be  represented  by  F  =  $  x  d  h  y.  Hence,  in  the  case  in  question 

' 


Hence  we  have  the  following  formula : 
— •  (**2  —  d,  }  h  y  -}-  R  =  <p  ft 


or,  simplified: 


If,  on  the  other  hand,  the  valve  has  to  rise  from  its  lowest  posi- 
tion after  the  water  has  been  cut  off,  then  the  excess,  or  the  differ- 


ADJUSTMENT  OF  THE  VALVES.  333 

ence  of  the  water  pressure  on  S  alone,  must  overcome  the  weight 
of  the  valve  system,  and  the  friction  of  the  pistons;  because  then 
the  pressures  on  both  sides  of  Gr  cease,  we  must,  therefore,  have 

|  d*(\  —  A2)y  =  R  +  *  *  (d, 
or,  more  simply  : 


These  formulas  will  serve  for  calculating  the  diameters  eZ,  and  d2 
of  the  two  pistons.  Neglecting  E,  which,  in  considerable  falls,  is 
almost  always  of  small  amount: 

d*  —  d*  =  4  *  (^  +  <22),  and  d  ^  =  4  *  K  +  d2),  therefore, 

f  d22  —  d*=d*,  or^22=2^2, 
and,  hence,  the  diameter  of  the  counter  piston  : 


,, 

or  about  f  of  the  diameter  of  the  piston  valve,  which  is  determined 
by  the  first  equation: 

d*-d?  =  4  1  (d,  +  dj,  or  d,  -  d  =  4  1, 
if  we  substitute  in  it  :  ^  >/2  for  dy 
We  then  have: 


.  4  1  =  2,414  .  4  »,  and  d2  =  3,414  .  4  *. 
x      — 

Taking  the  weight  of  the  pistons  into  account,  we  have,  with  suf- 
ficient accuracy, 


T         x 
and  from  this,  we  have  by  the  equation  1: 

d,  —  dl  =  4* 


For  the  sake  of  certainty  in  the  working,  both  diameters  are 
made  somewhat  greater,  and  the  excess  of  power  is  absorbed  by 
setting  the  regulating  cocks,  already  mentioned,  so  as  to  exactly 
adjust  the  area  of  passage.  Judging  from  the  best  existing  engines, 
we  may  take  4  1  =  0,1,  or  $  =  ^  In  order  that  in  the  passage 
of  the  pressure  water  through  the  valve  cylinder,  there  may  be  the 
least  possible  hydraulic  resistance,  it  is  usually  made  of  equal  area, 
at  that  part,  with  the  area  of  the  pressure  and  intermediate  pipes; 
and  supposing  the  formulas  give  a  diameter  dv  which  is  less  than 


334  ADJUSTMEXT  OF  THE  VALVES. 

that  of  the  pressure  pipes,  we  may  consider  that  there  exists  an 
excess  of  power,  which  must  be  adjusted  by  the  regulating  cocks. 

Example.  It  is  required  to  determine  the  proportions  of  a  two-piston  valve  system  for 
a  water-pressure  engine  of  400  feet  fall.     Suppose  the  weight  of  the  pistons  and  rod, 
&c.  =  150  Ibs.    Leaving  this  weight  out  of  the  calculation,  the  diameter 
d,  =  2,414  .  4  f  =  2,414  .  0,1  =  0,2414  feet  =  2,897  inches,  and  d3  =  3,414  .  0,1 
=  0,3414  =  4,097  inches.    Taking  the  weight  of  pistons,  &c.,  into  account  rf,  =  0,2414 

-I °'586  '  15° =  0,2414+ M!l=0,2414+ 0,0223  =  0,2637  ft.  =  3,164  inches, 

^0,05.400.62,5*  ^8,33.w 

and  d,  =  0,3414-l 0,243  .  150 — __0)34 14 +0,0092  =0,3516  feet  =  4,219  inches. 

It  will  be  sufficient  in  this  case,  if  we  take  rf,  =  3  J,  and  rf2  =  5  inches.  For  so  small  a 
counter-balance  to  piston  valve,  only  a  small  supply  of  water  is  necessary;  but  the  re- 
sistance in  the  passage  through  the  valve  cylinder  would  be  great.  If,  on  this  account, 
we  put  rf,  =  6  inches,  then  we  should  have  to  make  rfa  at  least  =  rf,  ^/2  =  8,484 
inches,  that  is  from  8|  to  9  inches,  the  excess  of  power  being  absorbed  by  adjusting  the 
cocks. 

§  201.  In  the  three-piston  valve  system,  the  mode  of  calculation 
is  very  similar  to  that  gone  through  above.  The  advantage  of  this 
system  is,  that  we  may  make  one  of  the  pistons,  the  valve  piston 
proper,  for  example,  of  the  same  diameter  as  the  pressure  pipes. 
The  calculations  for  the  valves  in  the  engine  represented  in  Fig. 
311,  may  be  made  as  follows :  Putting  dl  =  the  diameter  of  the 
lower  piston,  or  first  valve  piston,  and  d2  that  of  the  second,  and  d3 
that  of  the  upper  or  counter  piston;  then,  for  the  descent,  we  have 

i  \j*      ^2.^2  ,    4-#       A    u    ,   j    ,   j\ 
JL.J  04  —  «2  +  az  -| —  =  *t  $>  ^aa  +  a2  +  a3j, 

ft  h  y 

and  for  the  ascent : 

2.)  d*-d?-±*L  =  4  *  (d,  +  d,  +  dj. 

A  hy 

From  dl  we  can,  by  means  of  these  formulas,  determine  d3  and  d3, 
making  d2,  however,  somewhat  greater  than  the  calculation  gives 
for  insuring  certainty  of  action.  If  we  put  the  value  thus  found 
into  the  formula 


y 

we  get  as  the  diameter  of  the  third  piston : 


n  fly 

which,  for  the  reasons  already  given,  should  be  made  something 
more  than  the  absolute  result  of  calculation. 

For  the  valve  system  of  the  engine  in  Fig.  312,  we  have  the  fol- 
lowing formulas.  Let  A:  =  the  mean  height  of  the  pressure  column, 
and  A2  =  the  mean  height  of  counter-balance  column;  dl  the  diame- 
ter of  the  valve  piston,  d2  that  of  the  counter  piston,  and  d3  that  of 
the  projection  forming  a  third  piston.  The  power  in  the  descent, 
is  then 


ADJUSTMENT  OF  THE  VALVES. 

and  that  in  the  ascent  : 


therefore  : 


335 


If  dl  be  given,  we  can  then  calculate  d2  and  d3,  but  we  must  keep 
d2  somewhat  above,  and  d3  somewhat  below  the  result  of  the  formula. 
The  formulas 

1.  d*  —  #-8  **+  ««+  <*     and 


, 

are  of  rather  simpler  application. 

For  the  valve  system  shown  in  Fig.  317,  already  mentioned  as 
that  of  the  Clausthal  engines,  we  have,  when  dr  =  the  diameter  of 
valve  piston,  d2  the  diameter  of  upper  or  counter  piston,  and  da  that 
of  the  lower  or  auxiliary  piston,  the  power  for  descent  : 

Fig.  317. 


^     -f' 


336  WATER  FOR  THE  VALVES. 

The  power  of  ascent : 

—  [d3*  (Ax  —  A2)  —  d*  (hi  —  A2)  +  d22  AJ  y  —  R ;  therefore, 

1.)  d* -1  d*  -\ =  4  t  (dl  +  c?2  +  c?3),  and 

A.  «  n  y 


Example.  Supposing,  as  in  the  last-mentioned  engine,  A,  =  688  feet,  and  Aa  =  76  feet, 
R  =  170  Ibs.,  and  dl  =  i  foot,  we  get  the  diameters  of  the  other  pistons  as  follows : 

rfs'  =  8  *  (rf,  +  d2  +  rf3),  and  also  =  2  d,1  —  —'  d?  -f  8  R  ,  or,  in  numbers: 

h  ithy 

rfj"  =  0,2  (0,5  +  d2  +  rf3),  and  =  U,5  —  2,248  d* -f-  0,0107.  If,  now,  we  assume  d,  =  0,3 
feet,  we  have  by  one  formula  rf3a  =  0,5107 —  0,2023  =  0,3084,  that  is  d3  =  0,555; 
and  by  the  second  formula,  rf3a  =  0,2  .  1,355  =  0,2710,  t.  e,  d3  =0,5205.  But  if  we 
put  dz  =  0.33,  then  df  =  0,5107  —  0,2448  =  0,2659,  or  rf3  =  0,516,  and,  again, 
dsa  =  0.2  .  1,346  =  0,2692,  or  d3  =  0,519.  Hence  d,  =  0,33  .  12  =  3,96,  or  about  4 
inches,  and  d3  =  0,52  .  12=6,24,  or  6£  inches.  Jordan,  the  engineer,  who  erected 
these  machines,  has  made  4>:=  4  inches,  1,6  lines,  and  d3  =  5  inches,  9J  lines,  from 
which  we  deduce  that  4  ^  is  somewhat  less  than  0,1  in  this  case. 

Remark.  To  calculate  more  accurately,  the  diameter  of  the  valve  rod  would  have  to  be 
taken  into  account. 

§  202.  Water  for  the  Valves. — The  quantity  of  water  required, 
for  the  motion  of  the  valves,  gives  rise  to  the  loss  of  a  certain  amount 
of  mechanical  effect,  or  to  a  diminution  of  the  engine's  efficiency, 
because  it  is  abstracted  from  the  water  working  the  engine.  It 
should,  therefore,  be  rendered  as  little  as  possible,  that  is  d3,  the 
diameter  of  the  counter  piston,  and  its  stroke  should  be  as  small  as 
possible.  The  stroke  depends  on  the  depth  of  the  valve  piston,  or 
on  the  diameter  of  the  intermediate  pipe.  The  intermediate  pipe  is, 
therefore,  made  rectangular :  of  the  width  of  the  working  cylinder, 
and  low  in  proportion.  As  it  is  made  of  the  same  area  as  the  pres- 
sure pipes,  we  have  ad  =  ?— ^ ,  and,  therefore,  the  height,  or  least 

dimension  of  the  intermediate  pipe  a  =  ^— i-. 

4  (t 

That  the  valve  piston  may  cut  off  the  water  exactly  at  the  end 
of  the  stroke,  it  is  made  three  times  the  height  of  the  pipe,  or  its 
depth  is  al  =  3  a ;  and,  hence,  the  stroke  of  the  valve  piston  proper 
«1=a1  +  a  =  3a+a  =  4a,  and  the  quantity  of  water  expended 

for  each  stroke  is  =  ^— L  sl  =  ft  a  d*. 

If  the  engine  makes  n  strokes  per  minute,  the  quantity  of  water 
expended  by  the  valves  per  second. 

n       ns.      ft  d*      na      •,  2 

^  60  •-4^=60'"'" 
and,  hence,  the  loss  of  effect  corresponding : 


EXPERIMENTAL  RESULTS.  337 


or  the  loss  is  the  less  the  longer  the  stroke  of  the  engine. 

As  to  the  valve  gear,  the  power  required  to  work  it  is  so  small 
that  it  may  be  left  out  of  consideration.  The  study  of  the  arrange- 
ment of  the  mechanism  arises  under  another  section  of  our  work. 

Example.  If  in  the  water-pressure  engine,  the  subject  of  the  example  calculated 
there  be_  applied   a  valve  piston  of  9  inches  diameter,  and,  therefore,  a  counter  piston 
=  9^/2=13  inches  diameter.     If,  further,  the  intermediate  pipe  have  a  height 
«  =  -fj-  =4    ^  =  -gj-  =3,18  inches,  then  the  valve  piston  must  have  a  height 
a,  =  3  a  =  9,54  inches,  and  its  stroke  «,  =  a,  -f-  a  =  12,72  inches  =  1,06  feet  ;  and 
therefore  the  quantity  of  water  expended  each  stroke  =*  (—  Y  .  1,06  =  0,977  cubic 
feet  ;  and  hence  the  loss  of  effect  per  second  : 
L.  =  JL  .  0,977  .  h?  =  ±  .  0,977  .  350  .  62  .  5  =  1424.«feet  Ibs.,  or  nearly  3  horse  power. 

It  would  certainly  be  better  in  this  case  to  make  the  piston  valves  less  in  diameter,  and 
have  a  lower  intermediate  pipe;  for  although  this  would  increase  the  hydraulic  resist- 
ances,  still  it  would  not  involve  so  great  a  loss  as  the  waste  of  water  we  have  calculated 
implies. 

§  203.  Experimental  Results.  —  There  are  not  .many  good  experi- 
ments on  the  effect  of  water-pressure  engines.  These  engines  are 
usually  employed  as  pumping  engines  in  mines,  and  the  experiments 
k  -  that  have  been  made  involve  the  whole  machinery,  as  well  as  the 
>  engines  themselves,  in  the  results  as  to  the  efficiency.  But  it  is  very 
easy  to  get  an  approximate  determination  of  this  efficiency,  if  we 
assume  that  the  efficiency  of  water-pressure  engines  and  pumps  are 
in  certain  proportions  to  each  other.  This  assumption  we  may  make 
with  perfect  propriety,  as  the  engine  and  machine  are  very  analogous 
in  their  construction  and  movements.  We  shall  not  give  any  ad- 
vantage to  the  water-pressure  engine,  nor  be  far  from  the  truth,  if 
we  suppose  the  loss  of  effect  of  the  whole  apparatus  to  be  one-half 
due  to  the  water-pressure  engine.  The  calculation  then  becomes 
very  simple. 

The  effect  at  disposition  is  -^-  (Fs  4-  F&)  h  y,  in  which  J\  is  the 
section,  and  sl  the  stroke  of  the  auxiliary  piston.  The  effect  pro- 

duced, however,  is  ^F2h2y',  if  F2  =  the  section  of  the  pump  piston, 

60     ' 

and  hz  the  height,  the  water  is  raised  by  the  pump.  The  loss  of 
effect  is,  therefore, 

=  i(-F« 
the  half  of  which  is  : 


and  hence  the  efficiency  of  the  water-pressure  engine 


VOL.  ii.—  29 


338  EXPERIMENTAL  RESULTS. 

if  ijj  be  the  efficiency  of  the  combined  engine  and  pumps.  In  this 
mode  of  calculation,  it  is  assumed  that  there  are  no  losses  of  water, 
and  when  the  machinery  is  in  good  order,  this  loss  is  so  small  that 
it  may  be  neglected.  Jordan  found  for  the  Clausthal  engines,  that 
the  loss  of  water  in  the  water-pressure  engines  is  only  ^  per  cent., 
and  in  the  pumps  2  J  per  cent.  The  experiments  are  made  by  open- 
ing the  regulating  apparatus  in  pressure  and  discharge  pipes,  and 
then  raising  the  height  of  the  pump  column,  or  increasing  the  work 
to  be  done  till  the  required  number  of  strokes  is  performed  uniformly. 
By  experiments  on  this  principle,  Jordan  found  that  one  of  the 
Clausthal  engines  gave,  when  making  4  strokes  per  minute, 
^  =  0,6568,  and  making  3  strokes  «?j  =  0,7055,  and,  therefore,  in 

the  first  case,  »?  =  -^-  —  =  0,8284,  and  in  the  second  : 

rt  =  1'T°55  =  0,8527,  and  hence  as  a  mean  ,  =  0,84.     When  the 

greatest  effect  of  a  water-pressure  engine  cannot  be  determined  by 
the  method  of  heightening  the  pump  column  till  a  uniform  motion 
is  established,  it  may  be  done  perhaps  by  diminishing  the  water- 
pressure  column.  This,  however,  can  only  be  done  when  the  excess 
of  power  of  the  engine  is  small,  that  is,  when  the  part  of  the  water- 
column  to  be  taken  off  is  small.  The  water  may  be  kept  at  a  cer- 
tain level  in  the  pressure  pipes,  ascertained  by  a  float,  and  in  this 
way  the  efficiency  for  a  certain  head  be  determined.  The  engine  in 
Alte  Mordgrube,  near  Freyberg,  was  experimented  on  in  this  way, 
and  it  was  found  that  for  3  strokes  per  minute  ^  =  0,684,  and  hence 
the  efficiency  of  the  water-pressure  engine  alone  is  to  be  estimated 

as  ,  =  i??l  =  0,842. 

2 

The  most  of  the  results  reported  in  reference  to  the  effect  of 
water-pressure  engines  are  too  uncertain  to  be  worthy  of  much  con- 
fidence, having  been  deduced  from  experiments  in  which  essential 
circumstances  were  not  noted.  If  we  take  £  as  the  co-efficient  of 
resistance  corresponding  to  a  certain  position  of  the  regulating  valves 
or  cocks,  as  given  in  the  table,  Vol.  I.  §  340,  the  fall  y,  lost  by  this 
contraction,  may  be  estimated  by  the  formula  : 


and  we  can,  therefore,  estimate  the  efficiency  by  the  formula  : 
?&  _  V 


Example.  A  pressure  engine  consumes  10  cubic  feet  of  water  per  second,  besides  0,4 
cubic  feet  for  the  valves.  The  fall  =  300  feet,  the  mean  velocity  of  the  water  in  the 
pressure  pipes  =  6  feet  per  second,  and  the  circular  throttle  valves  in  the  main  pipe 
stands  at  60°.  Suppose,  that  by  this  engine,  there  is  raised  at  each  stroke  3,5  cubic  feet 
420  feet  high,  at  what  is  the  efficiency  of  the  engine  to  be  estimated?  According  to 
Vol.  I.  §  340,  for  the  position  of  the  valve  60°, 


CHAIN  WHEELS.  339 

118,  /.  £  .    l!=  118  .  0,0155  .  6'  =62,2  feet  ;  and  hence, 


§  204.  Chain  Wheels.  —  There  are  other  water-power  machines, 
neither  wheels  nor  pressure  engines,  but  which  are  to  be  met  with 
from  time  to  time.  We  may  mention  the  following  :  — 

The  chain  of  buckets  (Fr.  roue  a  piston  ;  Ger.  Kolbenrad)  has 
recently  been  revived  as  a  machine  recipient  of  water  power  by  La- 
molieres  (see  "Technologiste,"  Sept.  1845). 

The  principal  parts  of  this  machine  are  ACS,  Fig.  318,  over 
which  passes  a  chain  ADB,  form- 

ing the  axis  connecting  a  series  Fi?-  318- 

of  pistons  (called  buckets  or 
saucers),  E,  F,  G-,  &c.,  and  a  pipe 
EG,  through  which  the  chain 
passes  in  such  manner,  that  the 
pistons  nearly  fill  the  section  of 
the  pipe.  The  water  flowing  in 
at  _#,  descends  in  the  pipe  EGr, 
carrying  the  buckets  along  with 
it,  thus  setting  the  whole  chain  in 
motion,  and  turning  the  sprocket 
wheel  ACB  round  with  it.  La- 
moliere's  piston  wheel,  consists  of 
two  chains  having  from  10  to  15 
buckets  with  leather  packing. 
The  buckets  have  an  elliptical 
form,  the  major  axis  being  8  times 
the  minor  axis.  The  sprocket 
wheel  consists  of  two  discs  with 
six  cuts  to  receive  the  buckets. 

For  a  fall  of  two  metres  (6'  —  8"),  the  surface  of  buckets  being  0,25 
square  feet,  the  quantity  of  water  31  litres  (6,82  gallons)  per  second, 
the  number  of  revolutions  36  to  39,  it  is  said  that  an  efficiency  =  0,71 
to  0,72  was  obtained. 

Remark.  This  machine  is  the  chain  pump  used  in  the  English  navy,  converted  into  a 
recipient  of  power.  For  a  description  of  the  chain  pumps,  see  Nicholson's  "Operative 
Mech./'  p.  268. 

The  chain  of  buckets  (Fr.  noria,  chapelet,  pater-noster  ;  Ger. 
Eimerkette)  is  a  similar  apparatus.  The  chain  in  this  machine  has 
a  series  of  buckets  attached,  Fig.  319,  of  such  form  that  no  pipe  is 
required.  The  water  enters  at  A,  fills  the  buckets  successively, 
and  sets  the  whole  series  in  motion,  so  that  the  sprocket  wheel  C 
is  made  to  revolve.  This  wheel  should  give  a  very  high  efficiency, 
seeing  that  the  whole  fall  may  be  made  use  of;  but  from  the  great 
number  of  parts  of  which  it  is  composed,  their  liability  to  wear, 
and  other  sources  of  loss  of  effect,  it  is  practically  a  very  inefficient 
machine. 


340 


CHAIN  WHEELS. 


Fig.  319. 


Fig.  320. 


Remark.  We  may  here  mention,  that  the  so-called  rotary  pump,  rotary  steam  engines, 
&c.,  may  be  adapted  to  receive  water  power.  Fig.  320,  represents  a  water-pressure 
wheel,  of  which  there  is  a  detailed  description  and  theory  given  in  the  "  Polytechn. 
.Centralblatt.  1840."  It  is  Pecqueur's  rotary  steam  engine  adapted  to  water  power. 
BOB,  is  a  strong,  accurately  turned  axis,  JL  and  ^,  being  two  wings  connected  with  it, 
and  which  serve  as  pistons.  These  piston?  are  enclosed  in  a  cover  DEDtEt,  in  which 
there  are  four  slides  moved  by  the  engine  itself,  and  performing  the  functions  of  valves. 
The  axis  is  bored  three  times  in  the  direction  of  its  length,  and  each  of  the  hollow  spaces 
has  a  lateral  communication  within  the  cover.  The  pressure  water  flows  through  the 
inner  bore  O,  enters  through  the  side  openings  C  and  Ct,  into  the,  in  other  respects, 
isolated  space  between  the  axis  and  the  cover  ;  presses  against  the  pistons^?  and  At,  and 
in  that  way  sets  the  axis  in  rotation.  That  the  rotation  may  not  be  interrupted  by  the 
slides,  they  must  always  recede  before  the  piston  comes  up  to  them;  and  on  the  other 
hand,  that  no  water  pressure  may  act  on  the  opposite  side  of  the  piston,  the  slides  must 
fall  back  instantly  on  the  piston  passing  them,  so  that  the  spaces  JIB E  and  JllBlEl,  are 
shut  off,  and  communicate  only  with  the  passages  B  and  J3t,  through  which  the  water 
is  discharged  when  it  has  done  its  work. 

Mr.  Armstrong,  of  Newcastle,  constructed  a  water-pressure  wheel  of  about  5  H.  P.,  in 
1841,  a  description  of  which  will  be  found  in  the  "  Mechanic's  Magazine,"  vol.  xxxii. 

Literature. — We  shall  conclude  by  some  account  of  the  literature  and  statistics  of 
water  pressure  engines.  Belidor,  in  the  "Architecture  Hydraulique,"  describes  a  water- 
pressure  engine  with  a  horizontal  working  cylinder;  and  mentions,  also,  that,  in  1731, 
MM.  Denisard  and  De  la  Duaille  had  constructed  a  water-pressure  engine.  But  this 
machine  had  only  9  feet  fall,  and  raised  about  j1^  part  of  the  weight  of  the  power  water 
to  a  height  of  32  feet.  It  appears  pretty  certain,  however,  that  the  water-pressure  engine 
was  employed  for  raising  water  from  mines,  first  by  Winterschmidt,  and  soon  afterwards 
by  H61I.  The  details  of  this  historical  fact  are  to  be  found  in  Basse's  "Betrachtung  der 
Winterschmidt  und  Holl'schen  Wassersiiulenmaschine,  &c.,  Freiberg,  1 804."  A  drawing 
and  description  of  Winterschmidt's  engine  is  given  in  Calvcir's  "  Historisch  chronolog. 


WINDMILLS.  341 

Nachricht,  &c.,  des  Maschinenwesens,  &c.,  auf  dem  Oberharze,  Braunschweig,  1763." 
Roll's  engine  is  described  in  Delius's  "  Introduction  to  Mining."  originally  published  at 
Vienna,  1773,  and  in  the  description  of  the  engines  erected  at  Schemnitz,  by  Poda,  pub- 
lished at  Prague,  1771. 

Smeaton  mentions  the  water-pressure  engine  in  1765,  as  an  old  invention,  improved 
by  Mr.  Westgarth,  of  Coalcleugh,  in  the  county  of  Northumberland,  at  which  time  several 
had  been  erected,  in  different  mines,  on  Mr.  Westgarth's  plan.  See  Smeaton's  "  Report,'5 
vol.  ii.  p.  96. 

Trevethick,  the  celebrated  Cornish  engineer,  also  invented  or  reproduced  the  water- 
pressure  engine;  and  erected  one,  still  at  work  in  the  Druid  copper  mine,  near  Truro, 
about  the  year  1793.  See  Nicholson's  "Operative  Mech." 

The  water  pressure  engine  is  now  in  use  in  nearly  every  mining  district  in  the  world. 
The  Bavarian  engineer,  Reichenbach,  greatly  improved  and  has  made  a  most  extensive 
application  of  this  power  for  raising  the  brine  to  the  boiling  establishments  in  the  Salz- 
bourg  district.  These  engines  have  never  been  accurately  described,  but  notices  of  them 
will  be  found  in  Langsdorfs  "  Maschinenkunde,"  in  Hachette's  "Traite  elementaire  des 
Machines."  and  in  Flachat's  "Traite  elementaire  de  Mecanique."  The  engines  erected 
by  Brendel,  in  Saxony,  are  described  in  Gerstner's  "  Mechanik,"  where  also  the  engines 
in  Karinthia  and  at  Bleiberg  are  described  in  detail.  The  water-pressure  engines  in  the 
Schemnitz  district  are  described  by  Schitko,  in  his  "Beitragen  zur  Bergbaukunde."  Jor- 
dan has  given  a  very  detailed  account  of  the  engines  at  Clausthal,  in  Karsten's  "  Archiv 
fur  Mineralogie,"  &c.,  b.  x.,  published  as  a  separate  work  by  Reimer,  of  Berlin.  Junker 
has  described  his  engines,  at  Huelgoat,  in  the  "  Annales  des  Mines,"  vol.  viii.  1835,  and 
the  description  is  published  as  a  separate  work,  by  Bachelier. 

No  description  of  the  engines  erected  by  Mr.  Deans,  of  Hexham,  has  been  published. 
They  are,  however,  simple  and  efficient.  The  engine  erected  by  him  at  Wanlockhead, 
in  Scotland,  in  1830  or  31,  having  the  fall-bob  for  working  the  valves,  is  one  of  the  largest, 
and  considered  very  efficient 

But  the  water-pressure  engine  erected  in  1842,  at  the  Alport  Mines,  near  Bakewell, 

in  Derbyshire,  and  several  others  on  nearly  the  same  model,  are  perhaps  the  most  per- 

»      feet  of  this  description  of  engine  hitherto  made.     These  engines  have  been  constructed 

'"'r   from  the  designs  of  Mr.  Darlington,  engineer  of  the  Alport  Mines,  under  Mr.  Taylor,  by 

the  Butterly  Iron  Company.     There  is  a  beautiful  model  of  the  first  erected  at  Alport, 

in  the  Museum  of  Economic  Geology,  but  no  description  of  it  has  yet  been  published. 

Its  arrangement — the  construction  of  its  parts — the  valves,  and  their  gear — are  each  of 

them  admirable  and  peculiar  to  this  engine,  though,  in  its  general  features,  it  resembles 

the  engines  of  Brundel,  and  Junker,  and  Jordan,  which  have  been  described.— TB. 


CHAPTER    VII. 

ON  WINDMILLS. 

§  205.  Windmills.—  The  atmospheric  currents  caused  by  a  local 
expansion  of  the  air  by  the  sun's  heat,  are  a  source  of  mechanical 
effect,  as  is  the  expansive  force  of  air  heated  artificially. 

The  machines,  recipients  of  this  wind  power,  are  windmills  (Ft. 
roues  d  vent;  Ger.  Windrader).     They  serve  to  convert  a  portion 
of  the  vis  viva  of  the  mass  of  air  in  motion  into  useful  effec  .     As  the 
direction  of  the  wind  is  more  or  less  homonta  ,  windmills  o 
wheels  usually  have  the  axis  nearly  horizontal,  that  is,  they  a 


concave  buclcets  or  sails  have  be 
erected.     The  force  of  the  wind  against  a  hollow  surface 


342  WINDMILL  SAILS. 

than  against  a  plane  or  a  convex  surface,  and  hence  such  a  wheel 
revolves  under  very  light  winds,  but  not  advantageously  for  the  pro- 
duction of  mechanical  effect. 

Remark.  For  some  account  of  Beatson's  horizontal  windmills,  see  Nicholson's  "  Practical 
Mechanic,"  and  Gregore's  "  Mechanic,"  vol.  ii. 

§  206.  The  advantage  of  sail  wheels  over  any  construction  of 
bucket  wheel  is,  that  for  the  same  weight,  or  in  the  same  conditions 
generally,  they  produce  a  greater  effect  than  these  latter.  We 
shall,  therefore,  in  what  follows,  confine  ourselves  to  the  considera- 
tion of  sail  wheels,  of  which  the  general  arrangement  is  as  follows : 
First,  there  is  the  axle  of  wood,  or  better,  of  iron.  This  shaft,  or 
axle,  is  inclined  at  an  angle  of  from  5  to  15  degrees  to  the  horizon, 
in  order  that  the  wheels  may  hang  free  from  the  structure  on  which 
they  are  placed,  and  also  because  the  wind  is  supposed  to  blow  at 
an  inclination  amounting  to  that  number  of  degrees.  This  axle  has 
a  head,  a  neck,  a  spur  wheel,  and  a  pivot.  At  the  head  are  the 
arms — the  neck  is  the  journal,  or  principal  point  of  support  on  which 
it  revolves.  The  spur  wheel  transmits  the  motion  to  the  work  to  be 
done,  and  the  pivot,  at  the  low  end  of  the  axis,  takes  up  a  certain 
amount  of  the  weight  and  counter-pressure  of  the  machine.  The 
loss  of  .effect  arising  from  the  friction  of  the  axle  on  the  points  of 
support  is  considerable,  on  account  of  the  great  weight  and  strain 
upon  them,  as  also  on  account  of  the  velocity  with  which  it  generally 
revolves,  and  hence  every  means  must  be  taken  to  reduce  it.  On 
this  account,  iron  shafts  and  bearings  are  to  be  preferred  to  wooden 
ones,  as  they  may  be  made  of  much  less  diameter.  The  diameter 
of  a  wooden  neck  being  1^  to  2  feet;  that  of  an  iron  one  substituted, 
need  not  be  more  than  6  to  9  inches.  The  friction  of  wooden  axles 
is  also  in  itself  greater  than  that  of  iron. 

§  207.  Windmill  Sails. — A  windmill  sail  consists  of  the  arm  or 
whip,  of  the  cross  bars,  and  of  the  clothing.  The  whips  are  radial 
arms  of  any  required  length,  up  to  40  or  even  50  feet,  usually  about 
30  feet.  The  number  of  arms  is  generally  four,  less  frequently  5  or  6. 
For  30  feet  in  length,  these  arms  are  made  1  foot  thick  by  9  inches 
broad  at  the  shaft,  and  6  inches  by  4J  inches  at  the  outer  end. 
The  mode  Vrf  setting  them  in,  or  fastening  them  to  the  shaft,  is 
various.  When  the  axle  is  of  wood,  the  arms  are  put  through  two 
holes,  morticed  at  right  angles  to  each  other,  thus  getting  4  arms. 
The  arms  are  sometimes  made  fast  by  screws  to  the  shaft  head,  like 
the  arms  of  a  water  wheel,  and  we  refer  to  our  description  of  water 
wheels  for  hints  applicable  to  this  subject.  The  bars  are  wooden 
cross  arms,  passing  through  the  whip,  which  is  morticed  through  at 
intervals  of  from  15  to  18  inches  for  the  purpose,  at  right  angles  to 
the  leading  side  of  the  whip.  According  as  the  sail  is  to  have  a 
rectangular  or  a  trapezoidal  form,  the  bars  are  all  of  the  same  length, 
or  they  increase  in  length  from  the  shaft  outwards.  The  first  bar 
is  placed  at  ^  of  the  length  of  the  whip  from  the  shaft,  and  its  length 
is  =  to  this  I  to  |  of  the  length  of  the  whip.  The  outermost  bar  is 


POSTMILLS.  343 

made  from  '  to  §  of  the  length  of  the  whip.  The  whips  are  not 
generally  made  the  centre  line  of  the  sails,  but  they  divide  them  so 
that  the  part  next  the  wind  equals  from  £  to  \  of  the  entire  width 
of  the  sail.  Therefore,  the  bars  project  much  less  from  the  one  side 
than  from  the  other.  The  narrower  side  is  usually  covered  by  the 
so-called  windboard,  and,  on  the  wide  side,  the  winddoor  or  a  sail- 
cloth clothing  is  used. 

The  sails  are  made  plane,  or  surfaces  of  double  curvature,  i.  e., 
warped,  or  concave.  The  slightly  hollow  surfaces  of  double  curvature 
give  the  greatest  effect,  as  we  shall  learn  in  the  sequel.  For  plane 
sails,  the  bars  have  all  the  same  inclination  of  from  12  to  18  degrees 
to  the  plane  of  rotation.  In  the  double-curvature  sails,  the  first 
bars  are  set  at  24°,  and  the  outer  bars  at  6°  from  the  plane  of  rota- 
tion, and  the  inclinations  of  the  intermediate  bars  form  a  transition 
between  these  two  angles.  To  give  the  sails  concavity,  the  whips 
must  be  curved,  as  also  the  bars.  Although,  according  to  the  theory 
of  the  wind's  impulse,  this  form  gives  an  increased  effect,  the  diffi- 
culty of  execution  renders  it  nearly  inapplicable.  The  ends  of  the 
bars  are  connected  or  strung  together  by  uplongs,  and  sometimes 
there  are  3  of  these  uplongs  on  the  driving  and  2  on  the  leading  side 
of  the  sail,  to  strengthen  the  lattice  on  which  the  sailcloth  lies,  on  a 
series  of  frames  of  not  more  than  2  square  feet  each. 

§  208.  Postmills. — As  the  direction  of  the  wind  is  variable,  and 
the  axis  has  to  be  in  that  direction,  the  support  of  the  wheel  must 
have  a  motion  on  a  vertical  axis. 

According  to  the  manner  of  effecting  this  rotation,  windmills  may 
be  subdivided  into  two  classes:  the  postmill  (Fr.  moulin  ordinaire; 
Ger.  Bockmuhle],  Fig.  321,  and  the  smockmill,  or  towermill  (Fr. 
moulin  Hollandais;  Ger.  Hollandische,  or  Thurmmiihle). 

In  the  postmill,  the  whole  structure  turns  on  a  foot,  or  centre ; 
and  in  the  smockmill,  only  the  cap,  with  the  gudgeon  and  pivot  bear- 
ings resting  on  it,  turns. 

Fig.  321  is  a  general  view  of  a  postmill.  AA  is  the  post  or 
centre,  BB  and  BlBl  are  cross  bearers  or  sleepers,  framed  with 
struts  0  and  D,  to  support  the  post.  On  the  top  of  the  framing 
there  is  a  saddle  E.  The  mill  house  rests  on  two  cross  beams  FF, 
and  on  joists  GrGr,  as  also  on  the  cross  beam  If  on  the  head  of  the 
post,  which  is  fitted  with  a  pivot  to  facilitate  the  turning  of  the 
whole  fabric.  The  axis  turns  in  a  plumber  block  JV,  generally  of 
metal,  sometimes  of  stone  (basalt),  lying  on  the  beam  MM,  sup- 
ported on  the  framing  00.  KP,  KP,  &c.,  are  the  arms,  passing 
through  the  shaft  and  carrying  4  plane  sails  PP,  &c.  The  figure 
represents  a  grindingmill,  and,  hence,  the  wheel  transmitting  t 
power,  R,  works  into  a  pinion  Q,  driving  the  upper  millstone  b. 
In  order  to  turn  the  whole  house,  a  long  lever,  strongly  connected 
with  the  beams  EF,  projects  20  to  30  feet  from  the  back  of  it.  This 
lever  is  loaded,  to  counterbalance  the  weight  of  the  sail  wheel,  &c. 
When  the  mill  is  set  in  the  right  direction,  the  lever  .FT  (cut  off  in 


344 


POSTMILLS. 


the  figure),  is  anchored  and  held  fast,  and  generally  there  is  a 
movable  capstan  for  getting  power  to  turn  the  mill  house,  &c. 

Fig.  321. 


SMOCKMILLS. 


345 


§  209.  Smock-mills. — Smockmills  are  made  in  two  different  ways. 
Either  the  movable  cap  encloses  the  windshaft  alone,  or  a  greater 
part  of  the  mill  house,  from  the  windshaft  downwards,  turns  on  a 
vertical  axis.  The  motion  of  the  sail  wheel  is  transmitted  by  a  pair 
of  wheels  to  the  king  post,  that  is,  a  strong  vertical  axis  going 
through  the  whole  height  of  the  mill  house.  In  order  that  the 
wheels  may  be  in  gear  in  eve»y  position  of  the  windshaft,  it  is 
necessary  that  the  axis  of  the  one  shaft  should  intersect  that  of  the 
other. 

Fig.  322  represents  the  latter  arrangement,  which  is,  in  fact,  inter- 
mediate between  the  postmill  and  the  smockmill.  AA  is  a  etationary 

Fig.  322. 


tower  or  pyramid,  raised  above  which  is  the  building  containing 
the  machinery,  in  driving  which  the  power  is  consumed.  DD  is  the 
movable  top  of  the  mill,  supported  by  the  wooden  ring  Fl,  and  by 
the  wooden  ring  GG-,  by  means  of  the  uprights  EE  and  ££»  anc 
which  only  admits  of  rotation  round  these,  which  are,  in  fact,  the 
substitute  for  the  post  in  the  postmill.  The  mill  wheel  is  drawn 
round  by  a  capstan  R,  attached  to  the  lever  ff,  framed  by  the  stairs 
to  the  movable  part  of  the  structure.  The  windshaft  is  of  cast 


346 


SMOCKMILLS. 


iron,  and  rests  at  M  and  N  in  plumber  blocks,  lined  with  brasses. 
0  and  P  are  iron-toothed  wheels,  for  transmitting  the  motion  of  the 
wind  shaft  to  the  upright,  or  king  post  PP.  The  windsails  MS, 
US . . .  are  warped  surfaces  :  the  arms  are  fastened  by  screws  into 
the  cast  iron  socket  piece  R,  attached  by  wedges  to  the  head  of  the 
windshaft. 

The  upper  part  of  a  smockmill,  properly  so  called,  is  shown  in 
Fig.  323.     AA  is  the  upper  part  of  the  tower,  or  mill  house,  built 

Fig.  323. 


of  wood,  or  of  masonry.  BB  is  the  movable  cap,  ODE  is  the  wind 
shaft,  EE  the  arms  of  the  sails,  strengthened  by  the  ties  GrF,  GrF, 
supported  by  a  king  post  EGr.  JS'and  L  are  bevelled  gear  for  trans- 
mitting the  power  from  the  windshaft  to  the  vertical  shaft. 

The  sails  are  set  to  the  wind,  sometimes  by  means  of  a  lever,  as 
described  in  reference  to  the  last-mentioned  construction  of  cap,  but 
sometimes  by  means  of  a  large  wind  vane,  the  plane  of  which  is  in 
that  of  the  axis  of  the  wind  shaft,  but  more  generally  by  means  of 
a  small  windmill  S.  That  the  cap  may  revolve  easily,  it  is  placed 
on  rollers  c,  c,  c . . .  connected  together  in  a  frame,  and  running 
between  two  rings,  one  of  which  is  laid  on  the  summit  of  the  tower, 
the  other  is  attached  to  the  under  side  of  the  cap.  To  prevent  the 
cap  from  being  raised  up  and  displaced,  there  is  an  internal  ring  d, 
which  has  likewise  friction  rollers  running  on  the  internal  surface 
of  a  a.  When  this  method  of  adjustment  is  used,  the  outer  surface 
of  a  a  is  toothed,  and  a  small  pinion  e  working  into  it,  is  moved  by 
the  auxiliary  windmill,  by  means  of  the  bevelled  gear/  and  g,  Fig. 
323,  and  thus  the  whole  cap  is  made  to  revolve,  until  the  auxiliary 
wheel,  and  therefore  the  axis  of  the  windshaft  is  in  the  direction 
of  the  wind. 


REGULATION  OF  THE  POWER. 


347 


§  210.  Regulation  of  the  Power.— As  the  wind  varies  in  intensity 
as  well  as  in  direction,  when  the  work  to  be  done  is  a  constant  resist- 
ance, unless  some  means  of  regulating  the  power  be  applied,  the 
motion  of  the  machinery  would  not  be  uniform.  One  means  of  ab- 
sorbing any  excess  of  power,  is  a  friction  strap,  applied  to  the  out- 
side of  the  wheel  on  the  windshaft.  Another  means  is,  to  vary  the 
extent  of  sail,  or  the  quantity  of  clothing  exposed.  When  the  sails 
are  quite  spread  out,  the  maximum  power  depends  on  the  intensity 
of  the  wind,  and  if  this  intensity  be  constant,  the  power  may  be 
varied  by  taking  in  more  or  less  of  the  clothing  of  the  sail.  When 
the  clothing  is  canvas,  the  regulation  of  power  is  easily  managed  by 
reefing  more  or  less  of  it ;  and  when  the  clothing  consists  of  board- 
ing, the  removal  of  one  or  more  boards  answers  the  same  end. 

Self-adjusting  windsails,  that  is,  sails  which  extend  their  surface 
as  the  force  of  the  wind  decreases,  and  contract  it  as  this  force 
increases,  have  been  successfully  applied.  The  best  windsails  of 
this  kind  are  those  invented  by  Mr.  Cubitt,  in  1817,  and  of  which 

Fig.  324. 


- 


324  represents  the  section  of  a  part.     A  is  a  hollow  windshaft, 
a  rod  passing  through  it,  OD  a  ratchet  fastened  to  BL,  so 


348 


REGULATION  OF  THE  POWER. 


it  does  not  turn  with  it,  but  serves  to  move  it  in  the  direction  of 
the  axis. 

The  ratchet  works  into  a  toothed  wheel  E,  on  the  same  axis  as  the 
pulley  .F,  round  which  there  passes  a  string  with  a  weight  (r.  The 
sail  clothing  consists  of  a  series  of  boards,  or  sheet-iron  doors  be, 
JjCj,  &c.,  movable  by  the  arms  ac,  a^v  &c.,  round  the  axis  <?,  cv  &c. 
These  arms  are  connected  by  the  rods  ae,  atev  &c.,  and  by  the  levers 
or  cranks,  de,  d^v  with  toothed  wheels  d,  dL,  so  that,  by  the  turning 
of  the  latter,  the  opening  and  closing,  or,  in  general,  the  adjustment 
of  the  flaps  or  doors  is  possible. 

There  are  besides,  levers  BL,  BLV  Fig.  325,  revolving  on  centres 

Fig.  325. 


./Tand  Kv  and  attached  at  one  end  to  the  rod  BO,  and  at  the  other 
to  the  ratchets  LM  and  L^M^  working  into  the  small  wheels  d  and 
dr  The  drawing  explains  how  the  wind,  coming  in  the  direction  IF, 
works  backwards  on  the  counter-balance  weight  6r,  which  is  adjusted 
so  that  the  surface  exposed  shall  be  that  required  to  do  the  work 
regularly,  always  supposing  that,  for  the  maximum  surface  that  can 
be  exposed,  there  is  wind  sufficient. 

Remark.  Mr.  Bywater  invented  a  mode  of  furling  and  unfurling  the  clothing  when  it 
consists  of  sailcloth.     There  are  two  rollers  moved  by  toothed  wheels,  and  the  action  of 


DIRECTION  AND  INTENSITY  OK  T1IK  WIND. 


349 


these  is  to  cover  more  or  less  of  the  sail  frame,  according  10  the  force  of  the  wind.  This 
plan  is  described  in  detail  in  Barlow's  "Treatise  on  the  Manufactures  and  Machinery 
&c.  &c." 

§  211.  Direction  of  the  Wind.— The  direction  of  the  wind  may  be 
any  of  the  32  points  of  the  compass,  but  the  indications  are  gen- 
erally noted  as  one  of  the  8  following:  N.,  N.E.,  E.,  S.E.,  S.,  S.W., 
W.,  N.W.,  i.e.,  north,  north-east,  east,  south-east,  south,  south-west, 
west,  north-west;  or  naming  them  according  to  the  direction  from 
which  they  blow.  In  the  course  of  the  year,  the  direction  of  the 
wind  is  more  or  less  frequently  from  each  of  all  these  directions ; 
some  winds  blowing  more  frequently  than  others.  From  Kamtz's 
"  Meteorology,"  we  extract  the  following  table  of  the  winds  that 
blow  during  1000  days,  in  different  countries. 


Country. 

N. 

N.E. 

E. 

S.E. 

S. 

S.W. 

W 

N.W. 

Germany  . 
England   . 
France 

84 
82 
126 

98 
111 
140 

119 
99 
84 

87 
81 
76 

97 
111 
117 

185 
225 
192 

198 
171 
155 

131 
120 
110 

We  see  from  this  that,  in  the  three  countries  named,  the  south-west 
wind  predominates ;  the  passage  of  the  wind  from  one  direction  to 
another  is  usually  in  the  course  from  S.,  S.W.,  W.,  &c.,  and  seldom 
in  the  opposite  course  of  S.,  S.E.,  E.,  &c.  That  is,  the  latter  course 
is  generally  only  taken  through  a  small  angle,  and  then  retraced. 

The  wind  vane,  or  fane  (Fr.  girouette,  flouette ;  Ger.  Wind-  or 
Wetterfahne},  gives  the  direction  of  the  wind.  To  give  it  facility  of 
movement,  the  friction  on  its  pivot  or  collar  must  be  as  small  as  pos- 
sible, and  hence  the  blade  or  plane  of  the  vane  has  to  be  balanced 
by  a  counter-weight  to  bring  the  centre  of  gravity  line  to  pass  through 
the  axis  of  rotation.  (Whether  the  form  resulting  from  this  combi- 
nation gave  rise  to  the  term  weathercock  (Fr.  coq  a  vent;  Ger.  Wet'- 
terhahn),  or  whether  "a  king-fisher  hanging  by  the  bill,  converting 
the  breast  to  that  point  of  the  horizon  from  whence  the  wind  doth 
blow,  be  the  introducing  of  weathercocks,'"  we  cannot  pretend  to  say.) 

§  212.  Intensity  of  the  Wind. — The  miller  is,  however,  depend- 
ent on  the  intensity  of  the  wind,  and  not  on  its  direction ;  for  on 
the  former  the  mechanical  effect  to  be  obtained  from  given  wind- 
sails  depends. 

Accordingly,  the  velocity  of  the  wind  is 

Scarcely  sensible  for  1£  feet  per  second. 


Very  gentle  wind  for 
Gentle  breeze  for 
Brisk  breeze  for 
Good  breeze  for  windmills 
Brisk  gale  for 
High  wind  for 
Very  high  wind  for 
Storm  for 
Hurricane 
VOL.  ii. — 30 


6 

18 
22 
30 
45 
60 
70  to  90 

100  or  more. 


350  ANEMOMETERS. 

A  breeze  of  10  feet  per  second  is  not  in  general  sufficient  to  drive 
a  loaded  windsail,  and  if  the  velocity  rises  above  35  feet  per  second, 
the  intensity  becomes  too  much  for  the  strength  of  the  arms,  unless 
the  clothing  be  very  close  reefed,  and  stormy  weather  is  dangerous 
even  to  "bare  poles." 

Windgauges,  or  anemometers,  are  used  for  ascertaining  the  velo- 
city of  the  wind.  Many  anemometers  have  been  proposed  and 
adopted,  but  few  of  them  are  sufficiently  convenient  or  trustworthy 
in  their  indications.  The  anemometers  have  great  resemblance  to 
the  hydrometers  described  in  Vol.  I.  §  376.  The  velocity  of  a  cur- 
rent of  air  may  be  measured  by  noting  the  rate  of  progress  of  a 
body  floating  in  it,  as  a  feather,  smoke,  soap  bubbles,  small  air 
balloons,  &c.  This  means  will  not  suffice  in  the  case  of  high  velo- 
cities, for  the  eddies,  that  invariably  accompany  wind,  disturb  the 
progress  of  such  bodies. 

Anemometers  may  be  divided  into  three  classes.  Either  the  velo- 
city of  the  wind  is  deduced  from  that  of  a  wheel  moved  by  it,  or  it 
is  measured  by  the  height  of  a  column  of  fluid,  counterbalancing 
the  force  of  the  wind,  or  the  pressure  on  a  given  surface  is  deter- 
mined. We  shall  give  a  succinct  account  of  each  of  these  methods. 

Remark.  There  is  a  very  complete  treatise  on  Anemometers,  in  the  "  Allgemeirien 
Maschinenencyclopadie,  by  Hiilsse."  In  the  transactions  of  the  British  Association  for 
1846,  there  is  a  report,  by  Mr.  J.  Phillips,  on  Anemometers,  in  which  the  essential  points 
to  be  aimed  at  in  these  instruments,  and  the  merits  of  those  of  Whewell,  Osier,  and 
Lind,  respectively,  are  discussed.  The  chapters  on  Wind,  in  Kamtz's  "  Meteorology,"  and 
in  Gehler's  "  Worterbuch,"  are  standards  of  reference  on  this  subject. 

§  213.  Anemometers. — Woltmann's  wheel  may  be  used  for  ascer- 
taining the  velocity  of  the  wind  as  conveniently  as  it  is  for  ascer- 
taining the  velocity  of  currents  of  water.  When  its  axis  of  rotation 
is  set  in  the  direction  of  the  wind,  which  is  insured  by  means  of  a 
vane  set  on  the  same  vertical  axis  with  it,  the  number  of  revolutions 
made  in  a  given  time  may  be  observed,  and  from  this  the  velocity  may 
be  deduced,  as  explained,  Vol.  I.  §  378,  by  the  formula  v  =  v0  +  a  u, 
in  which  v0  is  the  velocity  of  the  wind,  for  which  the  wheel  begins 

to  stop,  and  o  is  the  ratio  v  V°.  If  the  impulse  of  the  wind  were 
of  the  same  nature  as  that  of  water,  and  if  they  both  increased 
exactly  as  the  squares  of  the  relative  velocities,  then  a  = 

would  answer  for  wind  and  water,  but  as  this  is  only  nearly  true, 
we  can  only  expect  that  the  co-efficient  a  is  nearly  the  same  for  wind 
and  water.  As  to  the  initial  velocity  v0,  this  is,  in  the  case  of  wind, 
about  v/800  =  28,3  times  greater  than  for  water,  as  the  density  of 
water  is  about  800  times  greater  than  that  of  air ;  and  thence  a 
column  of  air  800  times  as  high  as  that  of  water,  or  the  impact 
of  a  stream  of  ^/800  =  28,3  times  the  velocity  of  the  water.  This 
high  value  of  the  constant  v0,  makes  it  necessary  to  construct 
the  anemometer  sails  with  great  lightness,  and  to  have  the  axis  of 
hard  steel  running  on  agate  or  other  hard  bearings ;  as,  for  instance, 


ANEMOMETERS. 


351 


Fig.  326. 


in  Combe's  anemometer.  The  constants  v0  and  u  are  generally 
determined  by  moving  the  instrument  through  air  at  rest ;  but  this 
method  is  objectionable,  because  the  impact  of  a  fluid  in  motion 
is  not  the  same  as  the  resistance  of  a  fluid  at  rest  (Vol.  I.  §  391). 

It  is  better,  on  every  account,  to  deduce  the  constants  from  expe- 
riments on  currents  of  air,  deducing  the  low  velocities  by  direct 
observations  on  light,  floating  bodies.  By  placing  the  instrument  in 
the  main  pipe  of  a  blowing  engine,  the  observations  for  calculating 
the  constants  might  be  made.  The  calculation  of  constants  from 
a  series  of  experiments  for  which  v  and  u  are  known,  should  be  done 
as  shown  in  Vol.  I.  §  379. 

§  214.  Pitot's  tube  may  also  be  very  conveniently  applied  as  an 
anemometer.  This  is  Lind's  anemometer,  and  its  arrangement  is 
shown  in  Fig.  326.  AB  and  BE  are  two  upright 
glass  tubes  T'z  of  an  inch  in  diameter,  and  filled 
with  water,  and  BCD  is  a  narrow,  bent,  connect- 
ing piece  between  the  two,  of  only  ^  inch  diame- 
ter. FGr  is  a  scale,  by  which  to  read  oft*  the  height 
of  the  water.  When  the  mouth  A  is  turned  to 
the  wind,  its  force  presses  down  the  column  in 
AB,  and  raises  that  in  DE,  and  hence  the  dif- 
ference of  level  between  the  two  surfaces  may  be 
read  on  the  scale  h.  From  this,  the  velocity  of 
'-  the  wind  may  be  calculated  by  the  formula 

v  =  v0  +  a  v/£;  v0  and  a*  being  co-eflicients  de- 
duced from  experiments  for  each  instrument. 

The  use  of  this  instrument  is  very  limited,  as 
pressures  which  move  the  water  to  a  difference  of 
level  of  ^  of  an  inch  can  scarcely  be  noted  ac- 
curately, but  may  be  estimated  to  ^  or  5'n.  This  gives  5  to  7  miles 
per  hour  as  the  limit  of  wind  velocity  really  measurable.  To  obviate 
these  disadvantages,  and  render 

the  instrument  useful  for  mo-  Fig.  327. 

derate  velocities,  Robison  and 
Wollaston  introduced  the  fol- 
lowing improvements. 

In  Robison's  anemometer 
there  is  a  narrow,  horizontal 
pipe  HR,  Fig.  327,  between 
the  mouthpiece  A,  and  the  up- 
right pipe  BO',  and  there  is 
poured  as  much  water  into  the 
instrument,  before  using  it,  as 
brings  the  surface  F  to  the 
level  of  HR,  and  filling  the 
small  tube  to  H.  When  the 
mouth  A  is  turned  to  the  wind, 
the  water  is  driven  back  in  the  narrow  tube,  and  a  column 


352  ANEMOMETERS. 

counterbalancing  the  force  of  the  wind,  rises  in  the  tube  DE,  but 
which  is  measurable  by  the  length  of  tube  HHV  in  which  the  water 
has  been  driven  back.  If  d  and  dl  be  the  diameters,  and  h  and  hl 
the  height  of  the  columns  FF1  and  HHV  then 

!Lf?  h  =  !L*L.  hv  and  .-.  h  =    *.*  hv  or  h,  =  h  or  h,  is  always 


greater  in  the  ratio  (  —  )  than  A,  and  can,  therefore,  be  read  with 

1  d 

much  greater  accuracy  than  h.     If  —  =  5,  then  the  indications  in 

ffffl  are  25  times  greater  than  in  FFV 

Again,  the  differential  anemometer  of  Wollaston,  shown  in  Fig. 
328,  may  be  used  for  ascertaining  the  velocity  of  the  wind.     This 
instrument  consists  of  two  vessels  B  and  (7, 
Fig-  328-  and  of  a  bent  pipe  DEF.  which  unites  the 

two  vessels  by  their  bottoms.  The  one  ves- 
sel is  shut  at  top,  and  has  a  side  orifice  A, 
which  is  turned  to  the  wind.  The  instru- 
ment is  filled  with  water  and  oil.  The  former 
fills  the  two  legs  to  about  J,  and  the  oil  fills 
them  up,  and  occupies  part  of  each  of  the 
vessels.  The  force  of  the  wind  raises  the 
water  in  one  leg  higher  than  it  stands  in  the 
other,  and  the  amount  of  this  force  is  mea- 
sured by  the  difference  of  pressure  between 

the  water  column  FFV  and  the  oil  column  DDr  If  A  =  the  height 
of  each  of  these  columns,  and  «  =  the  specific  gravity  of  the  oil,  we 
then  have  in  the  last  formula  h  (1  —  *),  instead  of  A,  and,  therefore, 
v  =  r0  +  a  ^/(l  —  «)  h'.  If,  for  example,  the  oil  be  linseed,  *  =  0,94, 
v  =  Vo  +  a  ,7(1  —  0,94)  h  =  r0  +  a  ^0,06  .  h  =  v0  +  0,245  a  </h, 
or  A  is  'g0  =  16f  times  as  high  as  in  the  case  of  the  tubes  being 
simply  filled  with  water.  By  mixing,  or  combining  the  water  with 
alcohol,  the  density  of  the  water  may  be  brought  even  nearer  to  that 
of  the  oil  ;  and,  therefore,  1  —  «  becomes  still  less,  or  the  difference 
of  level  to  be  read  ;  and,  therefore,  the  accuracy  of  the  reading  is 
increased. 

§  215.  Anemometers,  analogous  to  the  stream  quadrant  (Vol.  I. 
§  381),  have  also  been  proposed  and  applied  on  the  same  principle, 
there  being  substituted  a  thin  plate  for  the  spherical  ball  used  in 
gauging  water  streams.  But  a  very  thin  metallic  sphere  is  certainly 
preferable  to  a  thin  plate,  for  then  the  force  of  the  wind  remains  the 
same  for  all  inclinations  of  the  rod  to  which  it  is  attached,  whereas  it 
changes  with  the  angle  of  inclination  of  the  thin  plate  ;  whilst,  when 
a  sphere  is  used,  the  formula  v  =  4-  \/tang.  ft  (in  which  /3  is  the 
deviation  of  the  rod  from  the  vertical),  is  sufficient.  The  application 
of  a  thin  plate  involves  a  complicated  expression  for  the  calculation 
of  the  velocity. 


FORCE  OF  WIND. 


353 


Lastly  the  velocity  of  the  wind  may  be  ascertained  by  the  force 
with  which  it  acts  directly  on  a  plane  surface  opposed  to  it  at  right 
angles,  and  for  this  the  instruments  used  are  more  or  less  similar 
to  the  hydrometer,  described  in  Vol.  I.  §  382.  If  the  law  of  the 
impact  of  wind  were  accurately  known,  the  velocity  of  the  wind 
might  be  determined  without  further  research  by  these  means.  But 
this  is  not  the  case,  and  the  formulas  given  in  Vol.  I.  §  390  and 
the  co-efficient  given  in  §  392,  lead  to  only  approximate  results. 
Retaining  these,  however,  for  the  present,  or  putting  the  impulse  of 
the  wind 


or,  rendered  in  English  measures,  as 

~  =  0,0155,  P  =  0,02883  v2  Fy, 

and  if  the  density  of  the  air  y  =  0,07974  Ib.  per  cubic  foot,  then 
P  =  0,002299  or  0,0023  v3  F,  and  v  for  two  square  feet  of  sur- 
face : 


P  =  0,0023  ........ 


20.85,/?  feet. 


f    » 
'<r 


For  velocities 

v  '= 

10 

15 

20 

25 

30 

35 

40 

45 

50 
feet 

The     impulsive 
force  of  the  wind 
on  1  square  ft.  = 

0,2455 

0,5524 

0.982 

1,534 

2,209 

3,007 

3,928 

4,971 

6.1375 
Ib. 

Fig.  329. 


Admitting  the  above  premises,  the  force  of  the  wind  on  any  sur- 
face at  right  angles  to  its  direction  may  be  easily  calculated. 

§  216.  Force  of  Wind. — We  shall  now  study  more  closely  the 
effect  of  the  impulse  of  wind  on  the  sails  of  windmills.  Let  us,  for 
this  purpose,  conceive  the  whole  sail  surface  divided  into  an  infinite 
number  of  normal  planes  on  the  axis 
of  the  sail  or  arm,  and  suppose  CD, 
Fig.  329,  to  be  such  an  elementary 
plane.  Owing  to  the  considerable  ex- 
tent, and  particularly  owing  to  the 
great  length  of  a  sail,  we  may  assume 
that  all  the  wind  of  the  column  press- 
ing on  the  surface  GD,  coming  in  the 
direction  AH,  will  be  turned  off  by 
the  impact  in  directions  parallel  to  GD, 
and,  therefore,  we  may  make  use  of  the 
formulas  in  Vol.  I.  §  388.  If  c  =  the 
velocity  of  the  sail,  Q  the  quantity  of  wind  striking  on  CD  per 
second,  y  =  the  density  of  the  wind,  and  o  =  the  angle  GAH,  which 
the  direction  of  the  wind  makes  with  GD,  then,  on  the  assumption 

30* 


354  BEST  ANGLE  OF  IMPULSE. 

that  the  plane  CD  moves  away  in  the  direction  of  the  wind,  the 

normal  impulse  of  the  wind  on  CD,  is  N=  e      v  sin.  a  .  Q  y. 

9 

Putting  the  section  ON=  Gr,  then  the  quantity  of  wind  Q  com- 
ing into  action,  is  not  Gr  c,  but  Gr  (c  —  v),  as  the  sail,  moving  with 
the  velocity  v,  leaves  a  space  G-  v  behind  it,  which  takes  up  a  propor- 
tion of  the  quantity  of  wind  G-  c  following  it,  equal  to  Gr  v,  without 
undergoing  any  change  of  direction.  Hence,  the  normal  impulse 
may  be  put 


-~  sin. 


9  9 

or,  if  F  =  the  area  of  the  element  CD,  and  we  substitute  F  sin.  a  for 

a,  then  N=  (c~v?  sin.  a2  FY. 

9 

Besides  this  impulse  on  the  face  of  CD,  there  is  a  counter  action 
on  the  back  ;  inasmuch  as  one  part  of  wind,  passing  in  the  directions 
CE  and  DF,  at  the  outside  of  the  plane,  takes  an  eddying  motion  to 
fill  up  the  space  behind,  and  consequently  loses  pressure  correspond- 
ing to  the  relative  velocity  (c  —  v  )  sin.  o,  and  represented  by 

/  -  _      \S 

-^  —  -  —  L  sin.  a2  .  Fy.     If  we  combine  the  two  effects,  we  get  the 
normal  impulse  of  the  wind  on  the  element  F  of  the  sail  : 


9 

217.  Best  Angle  of  Impulse.  —  In  the  application  of  this  formula 
to  windmills,  we  have  to  bear  in  mind  that 
Fig.  330.  the  windsail  BC,  Fig.  330,  does  not  move 

away  in  the  direction  AR  of  the  wind,  but 
in  a  direction  AP  at  right  angles  to  it, 
and  hence,  in  the  formula 

N=  3  .  (C—VY  sin.  a2  .  F7,  we  have  to  sub- 


stitute,  for  v,  the  velocity  Avl  =  v»  with 
which  the  windsail  moves  in  reference  to  the 
direction  of  the  wind.  If  v  =  the  actual 
velocity  of  rotation  A  v  (Fig.  330),  then 
Avl  =  Vj  =  v  cotg.  AvjV  =  v  cotg.  a  ;  and, 
therefore,  for  the  case  in  question  : 

F     or=  3  (c  sin'  a~v  c°8'  a?  Fr. 


'2g 

This  normal  impulse  is  to  be  decomposed  into  two  others,  P  and 
It,  one  acting  in  the  direction  of  rotation,  the  other  in  the  direction 
of  the  axis  of  the  element  of  the  sail  ;  then 


. 
R  =  N  sin.  .  =  3  ("'"•  --_ 


.  «  .  -Fy,  and 


BEST  ANGLE  OF  IMPULSE.  355 

By  multiplying  by  the  velocity  of  rotation  v,  we  get,  from  the 
formula  for  P,  the  mechanical  effect  of  the  windsail. 

T       r>        o  (c  *in-  «  —  v  cos-  a)2  -n 

L  =  Pv  =  3  i  -  -  -  L  v  cos.  a  .  F  y. 

2# 

The  parallel  or  axial  force  H,  gives  no  mechanical  effect,  but  on 
the  contrary  increases  the  pressure  on  the  pivot  or  footstep  at  the 
lower  end  of  the  windshaft,  and  so  gives  rise  to  a  loss  of  effect. 

The  last  formula  indicates,  and  it  is  self-evident,  that  the  effect 
increases  with  the  velocity  c,  and  with  the  area  F;  but  it  is  not  so 
evident  from  it,  how  the  angle  of  impulse  a,  affects  the  mecha- 
nical effect  produced.  That  L  may  not  be  =  0,  c  sin.  a.  must  be 

>  v  cos.  o;  that  is,  tana,  a  >  -,  and  cos.  a  >  0,    and,  therefore, 

c 
a  <  90°.     There  must,  therefore,  be  a  value  of  o  between  the  limits 

tang,  a  >  -,  and  a  <  90°,  corresponding  to  a  maximum  value  of  L. 

To  find  this  value,  let  us  instead  of  o  put  a  +^  x,  x  being  a  very  small 

angle.     Then  we  have  sin.  (o  -f  x)  =  sin.  a  cos.  x  -^  cos.  a  sin.  x,  or 

putting  cos.  x=\,  and  sin.  x  being  put  =  x, 

sin.  (a  •+_  x)  =  sin.  a  +_  x  cos.  a,  further: 

cos.  (a  4-  x}  =  cos.  a  cos.  x  -f  sin.  a  sin.  x  =  cos.  a  +  x  sin.  a,  and 

these  values  give  us  as  the  effect  : 

L  _  3g2y  FY  (  sin.  a—-  cos.  <*V  cos.  o, 
2g  V  c  I 

.'*+  ^  cos.  a^--(co 


=  ^L^_Fy[sin.a—-cos.*+  (cos.a  +  -sin.a.)xj(cos.a  +  xsin.  a) 
2g  c 

=  3(?2|)  F7  (Sin.  a  —  -  COS.  a)2  COS.  a 

+  [2  (Sin.  a  —  -  COS.  a)  (<?M.  a  +  ^  rin.  a)  COS.  a  —  (tin.  a  —  ^  «W.  a)2 

sz'n.  a]a;+),  &c.,  &c. 

=  L  +  —  Fy  ([2  («m.  a  —  -  C0«.  «*)  (cO»-  «  +  ^  ««»•  °)  ^'  ° 

_  («;n.  o  —  -  cos.  a)2  sin.  a.~]x  +  &c.) 

In  order  that  a  may  give  the  maximum  value,  L,  must  be  less 
than  L,*  being  increased  or  diminished  by  *,  that  is,  x  being 
positive  or  negative.  But  the  last  formula  gives  in  the  >  one  case 
\  >  L,  and  in  the  other  -  L,  so  long  as  the  second  member 
+  3  c  V  FY  [  ____  ]  x  is  a  real  quantity.  Therefore,  for  obtaining 
the  maximum  value,  it  is  necessary  that  this  second  member  should 
be  0,  or,  that 


356 


THE  MECHANICAL  EFFECT. 


2  (sin.  a  --  cos.  o)  (cos.  a  +  -  sin.  a)  cos.  a  —  (sin.  a  --  cos.  a}2  sin.  a=  0, 
c  c  c 

or  2  (cos.  a  -\  —  sin.  a)  cos.  a  ==  (sin.  a  --  cos.  a)  sin.  a, 


or  sn.  a2  --  «m.  a  cos.  a  =     00*.  a 


Dividing  by  cos.  a.2,  and  putting  — —  =  tang,  o,  we  have 

COS.  o 

tang,  o2 tang,  o  =  2, 

from  which  we  deduce  as  the  angle  for  the  maximum  effect: 

««*•— £  +  J(!;;)'+S- 

As,  in  the  windsails,  the  outer  elements  of  the  sail  have  a  greater 
velocity  than  those  nearer  the  axis  of  rotation,  it  follows  that  the 
outer  part  of  the  sail  should  be  set  at  a  greater  angle  of  impulse  than 
the  inner,  in  order  to  insure  a  maximum  effect.  Hence  the  sails 
must  not  be  plane  surfaces,  but  "surfaces  gauches,"  or  surfaces  of 
double  curvature,  warped  so  that  the  outer  part  deviates  less  from 
the  plane  of  the  axis  of  rotation,  than  the  inner  part. 

Remark.  The  most  advantageous  angles  of  impulse  of  a  sail,  may  also  be  ascertained 

by  the  following  construction.  Fig.  331, 
take  CB=  1,  set  off  CA  =  v/2  at  right 
angles  to  it,  i.  e.,  the  diagonal  of  a  square 
on  CB.  and  draw  AB.  The  tang.  ABC 
:=y/2,  and,  therefore, 
Z  ABC  =  54°:  44',  8",  which  is  the  angle 
of  impulse  close  up  to  the  axis  of  rotation. 

3  M  x 
If  in  y  =  _ ,  we  make  c  the  velocity 

of  the  wind,  and  a  the  angular  velocity, 
and  for  x  successively,  the  distance  of  the 
sail  bars  from  the  windshaft  axis,  and  set 
off  these  values  of  y  from  C  on  CB  as 
CD,,  CDp  CD3,  &c.  Further  draw  the 
hypothenuses  ADt,  AD%.  AD3,  &c.,  and 
prolong  them,  so  that  D,El  =  CDl.  -D3E2 
=  CD2,  D3E3  =  CD3.  &c.  Lastly,  lay 
off  AEr  AEV  AEy  in  the  direction  AC 
as  ACr  AC3,  ACy  &c.,  raise  at  C,,  C2,  C, 
&c.,  the  perpendiculars  C1BV  C^B^  (  3B3, 
&c.,  =  CB  =  I ,  and  draw  ABV  A±>2  Ab3, 
&c,  then  AB^C,,  AB2CV  AB3Cy  &c.,  are 
the  angles  required;  lor 


tang.  AB1C1  =  M '    '  =  ±fi  =  D,£,  +  AD,  =  y, 
tang.  AE,C,  =  ^L  =  ^  =  D>E,  +  AD, 


§  218.  T7ie  Mechanical  Effect.— The  formula  for  the  best  angle 
of  the  sails  may  be  used  conversely  to  determine  the  best  velocity  of 
rotation  for  a  given  angle  o.  For  this 


THE  MECHANICAL  EFFECT.  357 


tang,  a2  --  -  tang,  a  =  2, 

c 
and,  therefore,  very  simply, 


If  we  put  this  value  in  the  formula  for  the  mechanical  effect,  we 
have 
T       3  c2  j-,      tang,  a?—  2     c     (.  tang.J  —  2         \z 

Li  =  -^—  *  f  --  2  -  .  .        .   /  sin.  a  --  -^  --  COS.  a  \    COS.  a 

2g  tang,  a         3     \  3  tang,  a  / 

=  4     <?_  F      (tang,  a2  —  2)  cos,  a2  _  4    ^F      (3  *m.  a«  —  2) 
>'2<7      y>;  «n..»  5'2^  nn.  a3 

The  theoretical  effect  of  a  windsail,  may  hence  be  calculated  for 
any  given  velocity  of  wind,  and  of  rotation.  From  a  given  number 
of  revolutions  per  minute,  we  have  the  angular  velocity 

«  =  -^  =  0,1047  .  u.     If  the  whole  length  of  whip  be  divided  into 
oO 

7  equal  parts,  and  if,  as  usual,  the  sail  begins  at  the  lowest  point  of 
division,  so  that  its  total  length  =  f  ?,  we  can  very  easily,  by  means 
of  the  formula 


'  ..   calculate  the  best  angle  of  sail  a0,  uv  a2,  &c.,  or  for  each  of  the  points 
of  division  of  the  whip,  by  substituting  successively 

7  97  Q  ;  fj  I 

r0  =  w  •  ^'  vi  =  "  •  y  '  r»  =  "  •  y  —  to  ve  =  «  .  _  or  «  z. 

If,  further,  50,  lv  bt  ...  b6  be  the  width  of  sail  to  be  put  on  each 
of  these  points,  we  can  calculate,  by  aid  of  Simpson's  rule,  from 


«m.  a03      /         \       sin.  a.*  sin.  o2 

mean  value  &,  and,  hence,  we  arrive  at  the  whole  effect  of  the  sail 

L  =  $k  y  .  f  I  .  —  ,  or,  more  generally,  7,  being  the  length  of  sail, 

9  <? 

properly  so  called,  L  =  $  y  Jc  ^  —  . 

If  the  sail  were  a  plane  surface,  that  is,  if  a  were  constant  through- 
out its  whole  extent,  then,  by  means  of 

<*l  21   . 

v0  =  y>  vi  =  "-y»  &c>> 

we  should  first  calculate  the  corresponding  values  : 

^  COS.  a  .  &0,  (sin.  a  —  ^  C08.  aV^l  W«.  a  .  ft,,  &C. 
C  \  C  I     C 


J>  COS 
C 


and  then  from  these,  by  Simpson's  rule,  deduce  the  mean  value  ftp 
and  introduce  this  into  the  formula  for  the  mechanical  effect  dev»- 


loped,  L  = 

If  n  be  the  number  of  sails,  we  have  of  course  to  multiply  the 


358 


LOSS  OF  FRICTION. 


last  found  value  by  this  number,  to  get  the  whole  mechanical  effect 
developed  by  the  windsail  wheel,  or  L 


Example  1.  What  angle  of  impulse  is  required  for  a  windsail  wheel,  the  velocity  of 
the  wind  being  <!0  feet,  the  number  of  sails  4,  each  being  24  feet  in  length,  and  6  to  9 
feet  in  width?  Number  of  revolutions  16  per  minute.  What  will  be  the  theoretical 
effect  of  this  windmill? 

In  the  first  place,  the  angular  velocity  *>  =  0,1047  .  16  =  1,6755  feet,  and  if  the  dis- 
tance of  the  first  sail  bar  be  4  feet  from  the  axis  of  the  shaft,  or  the  total  length  of 
whip  =  24  +  4  =  28  feet,  then  for  the 


Distances  : 

4 

8 

12 

16 

20 

24 

28  Feet. 

The  velocities  : 
The  tangents  of  the  angles  of 
impulse  : 
The  angles  : 

The  values  »f  3  nn'  *"       2  • 

6,702 

2,004 
63°,29' 

0,5612 

6,0 
3,367 

13,404 

2,740 
69°,57' 

0,7810 

6,5 
5,076 

20,106 

3,575 
74°,22' 

0,8759 

7,0 
6,131 

26,808 

4,469 
77°,23' 

0,9220 

7,5 
6,915 

33,510 

5,397 
79°,30' 

0,9472 

8,0 
7,578 

40,212 

6,347 
81  °,3' 

0,9622 

8,5 
8,179 

46,914  ft. 

7,311 
82°,13' 

0,9716 

9,0  feet 
8,744 

sin.  «3 
The  width  of  sails  : 
The  product  of  the  two  last  : 

And  from  the  last  product  the  mean  value  : 

_  3,367  +  o,744  -f  4.  (5,076+  6,915  -f  8,179)  +  2  .  (6,131  +  7,578) 


12im  +  80,680+27,418       120,209 
18  18 


y  =  0,07974  Ibs.  f  /=  24,  and     .  =  0,0155  X  203  =  134, 

then  the  effect  of  this  wheel: 

Z  =  4  .  *  .  6,679  .  0,07974  .  24,124=  11,874  .  1,91  .  124  =  2798  feet  Ibs.  =  5  horse 

power. 

Example  2.  What  effect  may  be  expected  from  a  windmill  wheel,  having  four  plane 
sails,  and  the  angle  of  impulse  75°,  the  other  dimensions  and  proportions  being  the  same 
as  those  of  the  wheel  in  the  last  example  1  In  this  case 


The  velocities  of  ratio  —  : 
c 
The  differences 

0,3351 

0,6702 

1,0053 

1,3404 

1,6755 

2,0106 

2,3457 

V 

gin.  a,  cos.  a  : 
c 

0,8792 

0,7925 

0,7057 

0,6190 

0,5323 

0,4456 

0,3588 

The  width  b: 
The  products 

(sin.  a,  —  —  cos.  «)" 

6,0 

6,5 

7,0 

7,5 

8,0 

8,5 

9,0  feet 

X  -  cos.  a  b  : 
c 

0,4023 

0,7081 

0,9071 

0,9967 

0,9830 

0,8783 

0.7034 

From  the  latter  products  we  deduce,  by  Simpson's  rule,  the  mean  value 
k,  =  T>5  [0,4023  +  0,7034  +  4  (0,7081  +  0,9969  +  0,8783)  -f-  2  (0,9071+09830)] 

=  Tij  (1,1057+  10,3324+  3,7802)=  15'2183  _  0,8455,  and  from  this  we  have  the 

effect  required  Z  =  4  .  3  .  0,8455  .  0,7974  .  24  .  124  =  2390  =  4,34  horse  power,  in- 
stead of  5  horse  power,  found  when  the  sails  are  warped. 

§  219.  Loss  ly  Friction. — A  considerable  part  of  the  mechanical 
effect  developed  by  the  wind  on  the  sails,  is  consumed  by  the  fric- 
tion of  the  windshaft  at  the  neck,  especially  if  the  diameter  of  this 


LOSS  OF  FRICTION.  359 

be  great,  as  is  not  unfrequently  the  case.  We  may  assume  that 
the  whole  weight  of  the  sail  wheel  bears  on  the  neck,  and  Zs  leave 

TUT?  T  T V,0^  th*  Pressure  on  the  lower  or  back  bearing. 
Although  we  shall  thus  find  an  excess  of  friction,  yet  this  is  com- 
pensated by  leaving  out  of  consideration  the  friction  arisin^  On  the 
back  pivot  from  the  force  of  the  wind  in  the  axial  direction.  As 
the  back  pivot  is  much  less  in  diameter  than  the  neck  or  front  gud- 
geon, this  simplification  of  the  problem  may  be  the  more  readily 
admitted.  This  being  assumed,  we  have  from  the  weight  a  of  the 
whole  wheel,  F=*f  a  =  the  friction,  and  if  r  =  the  radius  of  the 
neck,  and  w  r  the  angular  velocity,  the  mechanical  effect  consumed : 


, 

if  v  be  the  velocity  at  the  periphery  of  the  sail  wheel. 

This  being  allowed,  flie  useful  effect  of  a  windmill  with  plane  sails: 


and  that  of  one  with  warped  sails  : 


From  the  formula: 

T       3  (c  sin.  a.  —  v  cos.  a)2  _ 

L  =  —> - L  v  COS.  a  .  Fy, 

for  the  theoretical  effect  of  an  element  of  a  sail,  we  may  deduce  the 
influence  of  the  velocity  of  the  sail  on  the  mechanical  effect,  and  we 

find  that  for  v  cos.  a.  =  c  s™'  a  (compare  Vol.  II.  §  118),  that  is, 

forv=      a™9'  a,  the  effect  is  a  maximum.     If  we  introduce  this 
o 

value  into  the  above  formula,  we  get 

-r         o        4        C3  Sin.  a3    p 

£  =  3.2V— fy—** 

and  from  this  we  deduce  that  the  effect  will  be  greatest  when  the 
angle  a  =  90°,  or  v  =  oo  .  These  conditions  cannot  be  fulfilled; 
because,  even  for  moderately  great  velocities,  the  prejudicial  resist- 
ances, and  more  particularly  the  friction  at  the  neck,  consume  so 
much  mechanical  effect,  that  the  useful  effect  remaining  is  very 
small.  The  velocity  of  rotation  should  be  great  to  insure  a  good 
efficiency,  but  it  must  in  each  case  be  made  a  special  subject  of 
calculation,  as  to  what  number  of  revolutions  will  give  the  maxi- 
mum effect.  This  can  only  be  done  by  calculating  the  effect  for  a 
series  of  velocities  of  rotation,  and  from  these  choosing  the  greatest, 
or  deducing  it  by  interpolation. 

Exampk.  Supposing  the  windshaft,  sails,  &c.,  of  the  mill  in  the  last  example  weighs 
7500  Ibs.,  that  the  radius  of  the  neck  or  gudgeon  r  =  i  foot,  that  the  co  efficient  of  friction 
/=0,1,  then  the  mechanical  effect  lost  by  friction  at  the  neck  =  0,1  .  7f>00  .  *r  =  4  19 
feet  Ibs.  There  remains,  therefore,  in  the  wheel  with  warped  sails  2798  —  4 1 9  =  2590 
feet  Ibs.,  or  about  86  per  cent,  of  the  theoretical  effect.  When  the  shaft  is  of  wood,  the 


360  EXPERIMENTS — SMEATON 's  MAXIMS. 

neck  is  double  the  above  diameter,  and,  hence,  the  loss  of  effect  by  friction  is  double,  or 
the  efficiency  is  only  0,70. 

§  220.  Experiments. — Experiments  or  observations  on  windmills, 
of  accuracy  sufficient  to  test  our  theory,  are  not  extant.  There  is 
no  lack  of  general  statements  of  the  results  of  the  effects  of  different 
windmills,  but  these  are  not  of  a  nature  to  serve  for  judging  of  the 
efficiency  of  the  machines  referred  to,  inasmuch  as  the  velocity  of 
the  wind  has  been  either  altogether  undetermined,  or  ascertained  by 
instruments  not  sufficiently  trustworthy.  The  experiments  of  Cou- 
lomb and  Smeaton  are  still  the  most  complete,  there  being,  in  fact, 
none  of  recent  date.  Coulomb  made  his  experiments  on  one  of  the 
many  windmills  in  the  neighborhood  of  Lille;  and  from  the  circum- 
stance of  the  work  done,  being  the  pressing  of  oil  by  means  of 
stampers,  a  kind  of  work,  the  mechanical  effect  consumed  in  which 
is  easily  calculated,  deductions  from  these  experiments  may  be  very 
safely  made.  The  four  sails  of  this  mill  were  warped  in  the  Dutch 
style,  with  the  angle  of  impulse  from  63|°  to  81^°,  and  each  of 
them  contained  about  20  square  metres,  or  215  square  feet.  The 
experiments  were  made  when  the  velocity  of  the  wind  was  from  7  to 
30  feet  per  second,  the  velocity  at  the  periphery  being  from  23  to  70 
feet,  and  the  results  correspond,  according  to  Coriolis  (see  "  Calcul 
de  1'effet  des  Machines),  with  those  of  the  theory  above  given.  It 
is,  besides,  easy  to  perceive  that,  for  the  better  construction,  when 

warped  sails  are  used,  the  mean  value  of — — .  cannot  vary 

sin.  a3 

very  much  from  that  which  is  deduced  by  calculation  in  the  first 
example  §  218,  viz.  =  0,880.  If,  now,  we  introduce  this  into  the 
general  formula,  we  obtain  the  following  very  simple  expression  for 
the  effect  of  a  windmill : 

L  =  |  .  0,88  .  0,0781  .nF—  =  0,000473  n  F  c3  ft.  Ibs. 

The  mean  of  Coulomb's  observations,  gives 

L  =  0,026  n  F  c3  kilogrammetres,  or 
£=  0,000511  wJV  ft.  Ibs. 

or  a  near  approximation  to  the  theoretical  determination.     We  may 

with  safety  assume 

L  =  0, 00048  nFc3  ft.  Ibs. 
This  formula  only  gives  satisfactory  results,  however,  when  the 

velocity  at  the  extremity  of  the  sails  is  about  2J  times  that  of  the 

wind,  as  indicated  by  theory  to  be  the  best  velocity. 

Example.  Suppose  a  windmill  of  4  horse  power,  when  the  velocity  of  the  wind  is  16 
feet  per  second  is  required.  What  sail  surface  must  it  have?  According  to  the  last 

formula,  n  F  = 4  -510      —  424932°  =  1030  square  feet;  that  is,  for  5  sails  each. 

0,00048  .  163  4(jy6 

206  square  feet     If/,  the  length  =  5  times  the  mean  breadth  b,  then 
5  6'  =  206  /.  6  =  v/IT  =  6£  feet,  and  the  length  /,  =  31$  feet 

§  221.  Smeaton  s  Maxims.  —  The  great  English  civil  engineer. 
John  Smeaton,  instituted  a  very  complete  inquiry  into  the  power  of 
wind,  and  made  a  series  of  experiments,  the  results  of  which  are 
given  in  the  following  table : — 


EXPERIMENTS  ON  WINDMILL  SAILS. 


361 


•jonpoij 
aqi  01  aoBjjng 

o> 

E* 

o  o 

odd 

•*!  °°.    .        .  t-  o  oo  t>  rj< 

r*  0»  CO         i-l  CO  T*  0  0  ^! 

00  <N  00  -• 

jo  O«;B^J 

0 

odd     d  d  d  d  d  d 

dodo 

do 

•lUnUIJXBttt  "&  JB 

PBCXJ  aqj  o;  pBoq 

0 

d 

CO  CO  rt 

oo"  oo  i> 

d  6i  d 

oq  to  01  o  T)<  r^ 
1    1    1      d  d  d  d  d  d 

d  d  d  d 

05 

JOOIJB^ 

"* 

B  IB  Xjiooja^Y 

d 

ilo 

t-  o      oo  oo  to 

III    o'§  !§§§ 

*->  CO  00  to 

to'  to  \fi  to 

dodo 

513 

jsgjuajS  jo  OIJBJJ 

•aoBjang 

i| 

sss 

SSS    oSSSSS 

O  O  O  O 

•*  to 

jo  luaixg; 

^ 

•* 

•jonpojj 

<x> 

CO 

rji  TT  n« 

(NOOt^         (NCOOOJ^-O 
O  —  I  Cl          TftOOOCOCOOO 
^CtO         ^OOtOtOO 

§>  C4  0>  (D 
I>  00  t»  t> 

O  to 

•pBO'J 

JSajE3jr) 

-2 

to  c*  ^ 

O  —  00 

i>'  oo  a 

J      1      I          CO  rt  rt  00  CO  O» 

111       «'  oo'  oo'  oi  d  o 

ssss 

CM  CO  TT  TC 

SI 

•uinuiiXBj\[ 

^ 

w«0 

O                   O                           OJ   rH 

oeoco       r-oocoto^ 

s§§§ 

^ 

~~ 

aqj  JB  sjieg 

o> 

o  as  o 

f-ei 
CO         CO  O>  00  t~  CO  O 

«„«« 

17 

aqj  jo  sujn^ 

•papBojun  sjiBg 

g 

,§s 

,    ,    i       o  o   ,  co  oo  o 

III     22  1  222 

co  r-  •*  to 

«  to 

0  04 

aqj  jo  siunj, 

•aSuBjsajBai 

to 

o»  o  oo 

tOCiJ^J         OOOOOiOt^ 

^Scico 

CM  CM 

•sapituauxa 

to 

O>  O  00 

oscjo      ocoiot»oo» 

HM 
t^  O  <N  O 

<N(N 

aqi  JB  aiSuy 

_ 

(N  CO  Tj" 

UJtOl^         OOOSO-^INCO 

•*  o  to  r- 

oo  r. 

The  description  of  sails  made 
use  of. 

Plane  sails,  at  an  angle  of  55°  .  . 

Plane  sails,  weathered  according  1 
to  the  common  practice  .  .  1 

Weathered  according  to  M'Laurin's  ) 
Theory  •  •  / 

Sails  weathered  in  the  Dutch  man-  J 
tier,  tried  in  various  positions  .  j 

Sails  weathered  in  the  Dutch  man-  f 
ner,  but  enlarged  towards  theV 

Eight  sails,  being  sectors  of  ellip-  C 
ses,  in  their  best  positions  .  £ 

VOL.  II.— 31 


362  SMEATON'S  MAXIMS. 

The  experimental  wheel  had  whips  21  inches  long,  the  sails  being 
18  inches  long,  and  5,6  inches  broad.  This  wheel  was  not  moved 
by  the  impulse  of  wind,  but  was  moved  round  in  air  at  rest,  whence 
it  was  the  resistance  of  the  air,  and  not  its  impulse,  which  was  ob- 
served— a  circumstance  taking  considerably  from  the  value  of  the 
experiments.  The  motion  of  the  sails  against  the  wind,  was  given 
by  means  of  an  upright  shaft,  from  which  projected  an  arm  5J  feet 
long,  at  the  end  of  which  was  a  seat  for  the  model  mill  wheel.  This 
upright  shaft  was  set  in  motion  by  the  observer  having  a  cord  wound 
round  it  like  the  peg  of  a  top.  To  measure  the  resistances  of  the 
air,  supposed  here  to  be  identical  with  the  impulse  of  wind  of  the 
same  velocity,  there  was  a  scale  with  weights,  attached  by  a  fine 
cord  to  the  shaft  of  the  wind  wheel,  and  this  was  wound  up  by  the 
power  communicated  to  the  sails.  The  results  of  these  experiments 
correspond  well  qualitatively  with  our  theory.  They  show  to  demon- 
stration that  the  warped  sail  gives  the  best  effect,  and  that  the 
angles  of  impulse  deduced  by  theory  are  actually  the  best.  In  the 
example  to  §  218,  we  found  the  angles  for  7  bars,  starting  from  next 
the  axle,  to  be:  63°  29';  69°  57';  74°  22';  77°  23';  79°  30';  81° 
3',  and  82°  13',  and  Smeaton  found  the  following  6  angles  to  be  the 
best,  or  at  least  very  good,  72° ;  71° ;  72° ;  74°  ;  77J°  ;  83° ;  or 
very  little  different  from  the  theory. 

Smeaton  remarks,  too,  that  a  deviation  of  2  degrees  in  the  angle  of 
impulse,  has  no  sensible  influence  on  the  mechanical  effect  produced 
by  the  wheel. 

Smeaton  draws  the  following  maxims  from  his  experiments,  made 
at  velocities  varying  from  4J  to  8f  feet  per  second. 

1.  The  velocity  of  the  windmill  sails,  whether  unloaded  or  loaded, 
so  as  to  produce  a  maximum,  is  nearly  as  the  velocity  of  the  wind, 
their  shape  and  motion  being  the  same. 

2.  The  load  at  the  maximum  is  nearly,  but  somewhat  less  than, 
as  the  square  of  the  velocity  of  the  wind,  the  shape  and  position  of 
the  sails  being  the  same. 

3.  The  effects  of  the  same  sails  at  a  maximum  are  nearly,  but 
somewhat  less  than,  as  the  cubes  of  the  velocity  of  the  wind. 

4.  The  load  of  the  same  sails  at  the  maximum  is  nearly  as  the 
squares,  and  their  effects  as  the  cubes  of  their  number  of  turns  in  a 
given  time. 

5.  When  the  sails  are  loaded  so  as  to  produce  a  maximum  at  a 
given  velocity,  and  the  velocity  of  the  wind  increases  the  load  con- 
taining the  same:  first,  the  increase  of  effect,  when  the  increase  of 
the  velocity  of  the  wind  is  smaller,  will  be  nearly  as  the  squares  of 
those  velocities;  secondly,  when  the  velocity  of  the  wind  is  double, 
the  effects  will  be  nearly  as  10  to  27 J;  but,  thirdly,  when  the  velo- 
cities compared  are  more  than  double  of  that  where  the  given  load 
produces  a  maximum,  the  effects  increase  nearly  in  a  simple  ratio 
of  the  velocity  of  the  wind. 

6.  If  sails  are  of  a  similar  figure  and  position,  the  number  of 


SMEATON'S  MAXIMS.  363 

turns  in  a  given  time  will  be  reciprocally  as  the  radius  or  length  of 
the  sail. 

7.  The  load  at  a  maximum  that  sails  of  a  similar  figure  and  posi- 
tion will  overcome,  at  a  given  distance  from  the  centre  of  motion, 
will  be  as  the  cube  of  the  radius. 

8.  The  effect  of  sails  of  similar  figure  and  position  are  as  the 
square  of  the  radius. 

9.  The  velocity  of  the  extremity  of  Dutch  sails,  as  well  as  of  the 
enlarged  sails,  in  all  their  usual  positions  when  unloaded,  or  even 
loaded  to  a  maximum,  is  considerably  quicker  than  the  velocity  of 
the  wind. 

According  to  these  experiments,  the  effect  of  the  wind  on  wind- 
mill sails  is  greater  than  theory  indicates,  or  than  Coulomb's  experi- 
ments gave.  j.,^ 

Literature.  The  most  complete  exposition  of  the  theory  of  windmills  is  given  in  Weis-  \ 
bach's  "  Bergmaschinen  Mechanik,"  vol.  ii.,  arid  in  Corioiis's  "  Traite  du  Calcul  a  1'effet 
des  Machines."  Smeaton's  experiments  are  recorded  in  the  "  Philosophical  Transactions," 
1759  to  1776.  They  were  collected  into  a  separate  volume,  and  published  under  the 
title  "An  experimental  Enquiry  concerning  the  natural  powers  of  Water  and  Wind  to 
turn  Mills  and  other  Machines  depending  on  a  circular  motion."  These  papers  were 
translated  into  French  by  Girard,  in  1827.  There  are  extracts  from  them  in  Barlow's 
"Treatise  on  the  Manufactures,"  &c.  In  Nicholson's  "Operative  Mechanic,"  Brewster's, 
Ferguson's,  &c.,  &c.  Coulomb's  experiments  are  given  in  his  oft-quoted  work  "Theorie 
des  Machines  simples." 

Mariotte  wrote  upon  the  impulse  of  wind,  in  his  "  Hydrostatics."  He  makes  the  impulse 


Borda,  in  the  "Memoires  de  1'Academie  de  Paris,"  1763,  has  a  paper;  Rouse,  Hutton, 
Woltmann,  have  all  handled  this  subject.  The  two  latter  authors  find  P  much  smaller 
than  Mariotte  did,  because  they  measured  the  resistance,  not  the  impulse  of  the  wind. 

The  co-efficient  £  s=  *,  as  found  by  Woltmann,  is  too  small,  or  P  =  *  —  F  y  is  certainly 

too  little,  for  he  did  not  obtain  the  constants  for  his  windsail  wheel  by  direct  experiment 
(see  "Theorie  und  Gebrauch  des  Hydrometrischen  Fliigels,"  Hamburg,  1790).  Hutton 
deduces  from  his  experiments,  that  it  is  more  accurate  to  consider  the  impulse  and  resist- 
ance of  the  air  as  increasing  as  F°-1  (see  "  Philosophical  and  Mathematical  Dictionary," 
vol  ii  )  If  we  assume  £  =  1,86  for  a  small  surface  of  1  square  foot,  then,  for  a  sail 
of  200  square  feet  surface,  we  should  have  £  =  2000-'  .  1,86  =  1,7  .  1,86  =  3,162,  which 
agrees  well  with  the  theoretical  determination,  and  with  what  we  have  said  above, 

where  ?  =  3  and  P  =  3  .  —  F  y.     In  Poncelet's  "  Introduction  &  la  Mecanique  indus- 

2g 

trielle,"  there  is  an  admirable  collection  and  discussion  of  the  experiments  on  impulse 
and  resistance  of  wind.  ^* 


INDEX. 


A. 

Abutting,  resistance  of  earth,  20 
Abutments,  stability  of,  31 

• piers,  &c.,  66 

Animals,  power  of,  121 

,  formulas  for,  123 

Arches,  27 

,  line  of  pressure  and  resistance  of, 

28 

,  loaded,  33 

,  stability  of  abutments  of,  31 

,  semi-circular   arch,  with   parallel 

vaulted  surfaces,  table  of,  37 

• ,  arches,  masonry  at  the 

back,  of  45°  inclination,  table,  37 

,  arches,  with  horizontal 


masonry  above,  table  of,  38 

,  tables  of,  36 

i  table  of  the  relative  dimensions  of 

J,  39 
-,  test  of  equilibration  of,  34 

,  vaulted  arches,  with  parallel  arched 

surfaces,  table  of,  38 
Axles  and  gudgeons,  170 

B. 

Backwater,  136 

,  amplitude  of,  144 

,  swell,  146 

Balance,  common,  105 

,  index  or  bent  lever,  115 

,  sensibility  of,  107 

,  spring,  116 
,  stability  and  motion  of,  108 

,  unequal  armed,  109 
Batter,  retaining  walls  with,  26 
Beams,  compound,  55 
Borda's  turbine,  234 
Braces  or  struts,  54 
Brake,  of  friction,  119 
Bridges,  chain  or  suspension  of,  59 

,  of  weigh,  111 

,  of  portable  weigh,  113 

,  timber,  48 

,  tubular,  82—95 

Bucket  wheels,  165 


Buckets,  form  of,  for  water  wheels.  174 

,  construction  of  the,  271,  290 

• ,  number  of,  for  water  wheels,  186 

Burdin's  turbines,  236 


Cadiat's  turbine,  247 

Capstan,  vertical,  131 

,  horse,  131 

Centrifugal  force,  effect  of,  188 

Chain,  or  suspension  bridges,  59 

Chains  and  ropes,  sectional  dimensions  of, 
63 

,  elongation  of,  65 

Chain  wheel,  339 

Cohesion  of  semi-fluids,  16 

Combe's  reaction  water  wheel,  246 

Compound  beams,  55 

Common  balance,  105 

Conduit  pipes,  158 

Conway  Tubular  Bridge,  82 

Corn  seeds,  semi-fluid,  1 3 

Coulomb's  experiments  on  the  rigidity  of 
hemp  ropes,  102 

Crushing,  Table  of  the  resistance  of  mate- 
rials to,  69 

Curb,  construction  of  the,  202 

wheels,  mechanical  effect  of,  204 

Cylinder,  the  working,  300 

,  engine  combined,  319 


D. 


Dams,  136 

Darlington's  water-pressure  engine,  319 
Dimensions  of  parts  of  water  wheels,  167 
Dykes,  149 

,  stability  of,  150 

,  ofliet  sluices  of,  152 

Dynamometer,  105,  116 


Earth,  semi-fluid,  13 

,  abutting  resistance  of,  20 

,  pressure  of,  14 

-,  surcharged  masses  of,  17 


368 


INDEX. 


Water  wheels  (VERTICAL),  formula  for  the 

total  effect,  206 
,  friction   of    the    gudgeons, 

total  effect,  191—192 

,  in  straight  courses,  213 

,  overfall  sluices,  198—200 

,  partition  of,  215 

,  penstock  sluices,  200 — 202 

,  Poncelet's,  218—225 

,  proportions  of    the,   172 — 


174 


214 


undershot  wheels,  211 
,  useful  effect  of  undershot, 


-,  useful  effect  of  wheel,  high- 
breast  wheel,  192—198 

-,  sluices,  pentroughs  of,  177 


—180 


-,  recent  experiments  on,  225 
-,  small,  227 


Weigh-bridges,  111 
Weirs,  construction  of,  137 

,  discontinuous,  142 

Wheels,  see  water  wheels. 
Wheel,  useful  effect  of,  192 


Wheel,  high-breast,  195 

,  chain,  339 

,  construction  of,  166 

,  the  proportions  of,  172 


Whitelaw's  turbines,  278 
Windlass,  the,  129 
Windmills,  341 

,  anemometers,  350 

,  best  angle  of  impulse,  354 

,  direction  of  the,  369 

,  experiments  on  the,  360 

,  force  of  the,  353 

,  intensity  of  the,  349 

,  loss  of  friction,  358 

,  mechanical  effects  of  the,  356 

,  postmills,  343 

,  regulation  of  the  power  of,  347 

,  sails,  342 

,  Smeaton's  maxims  on  the,  360 

,  smockmills,  343 

,  tables  of  nineteen  experiments 

on  the,  361 

,  vane,  349 

Woltmann's  wheel,  350 
Wooden  structures,  40 


THE  END. 


CATALOGUE 

OF 

BLANCHAED  &LEA'S  PUBLICATIONS. 

CAMPBELL'S  LORD  CHANCELLORS.    New  Edition-  (Just  I.sned  ) 

LIVES  OF  THE  LORD  CHANCELLORS 

KEEPERS  OP  THE  GREAT  SEAL  OF  ENGLAND, 

FROM  THE  EARLIEST  TIMES  TO  THE  REIGN  OF  KING  GEORGE  IV. 
BY  LORD  CHIEF-JUSTICE  CAMPBELL,  A.  M.,  F,  R,  S.  E, 

Second  American,  from  the  Third  London  Edition. 
Complete  in  seven  handsome  crown  8vo.  volumes,  extra  cloth,  or  half  morocco. 

This  has  been  reprinted  from  the  author's  most  recent  edition,  and  embraces 
his  extensive  modifications  and  additions.  It  will  therefore  be  found  eminently 
worthy  a  continuance  of  the  great  favor  with  which  it  has  hitherto  been  received. 

Of  the  solid  merit  of  the  work  our  judgment  may  be  gathered  from  what  ha*  already 
been  said.  We  will  add,  that  from  it*  infinite  fund  of  anecdote,  and  happy  variety  of 
style,  the  hook  addresses  ilself  with  equal  claims  to  the  mere  gen-™!  reader,  us  to  the 
legal  or  historical  inquirer;  and  while  we  avoid  the  stereoiyped  commonplace  of  affirra- 
iim  that  MO  lihrary  can  he  complete  wi.hout  it.  wf  feel  constrained  to  afford  it  a  higher 
tribute  hy  pronouncing  it  emitted  to  a  dismiguished  place  on  the  shelves  of  every  scholar 
who  i*  fortunate  enough  to  po**«-ss  it  —  Frazer's  Magazine. 

A  work  w,,ich  will  lake  its  place  in  our  libraries  as  one  of  the  most  brilliant  and 
valuable  contribution*  to  the  literature  of  the  present  day.— Athentrum. 

The  brilliant  success  of  this  work  in  England  is  hy  no"  means  greater  than  its  meriig. 
It  is  certainly  the  most  brilliant  contribution  to  English  history  made  within  our  recollec- 
tion ;  it  has  ihe  charm  and  freedom  of"  Biography  combined  with  the  elaborate  and  care- 
ful comprehensiveness  of  History .-— iV.  Y.  Tribune. 

BY  THE  SAME  AUTHOR— TO  MATCH— (Now  Ready.) 
LIVES  OF  THE 

CHIEF-JUSTICES  OF  ENGLAND, 

From  the  Norman  Conquest  to  the  Death  of  Lord  Mansfield. 

SECOND    EDITION. 

In  two  very  neat  vols.,  crown  8vo.,  extra  cloth,  or  half  morocco. 
To  match  the  "Lives  of  the  Chancellors"  of  the  same  author. 

In  this  work  the  author  has  displayed  the  same  patient  investigation  of  histo- 
rical facts,  depth  of  research,  and  quick  appreciation  of  character  which  have 
rendered  his  previous  volumes  so  deservedly  popular.  Though  the  "  Lives  of 
the  Chancellors"  embrace  a  long  line  of  illustrious  personages  intimately  con- 
nected with  the  history  of  England,  they  leave  something  still  to  be  filled  up  to 
complete  the  picture,  and  it  is  this  that  the  author  has  attempted  in  the  present 
work.  The  vast  amount  of  curious  personal  details  concerning  the  eminent 
men  whose  biographies  it  contains,  the  lively  sketches  of  interesting  periods 
of  history,  and  the  graphic  and  vivid  style  of  the  author,  render  it  a  work  of 
great  attraction  for  the  student  of  history  and  the  general  reader. 

Although  the  period  of  history  embraced  by  these  volumes  had  bern  previously  tra- 
versed by  the  recent  work  of  the  noble  and  learned  author,  and  a  great  portion  of  its 
most  exciting  incidents,  especially  those  of  a  constitutional  nature,  there  narrated,  yet 
in  uThe  Lives  of  the  Chief-Justices"  there  i<  a  fund  both  of  interesting  information  and 
valuable  matter,  which  renders  the  book  well  worthy  of  perusal  by  every  one  who 
desires  to  oo'ain  an  acquaintance  with  the  constitutional  history  of  his  country,  or  as- 
pires to  the  rank  of  either  a  statesman  or  a  lawyer.  F.  w  lawyers  of  Lord  C«m»l>«ir» 
eminence  could  have  produced  such  a  work  ag  he  has  put  forth.  None  hut  lawyers  of 
his  experience  and  acquirements  could  have  compiled  a  work  combining  the  same  in- 
terest as  a  narration,  to  (he  public  generally,  wiih  ihe  same  ninount  of  practical  infor- 
mation for  professional  aspirants  more  particularly.— Britannia. 


2    BLANCHARD  &  LEA'S  PUBLICATIONS.— (History  <$•  Biography.-) 
NEE3BUHR' S  ANCIENT  HISTORY— (A  new  work,  now  ready.) 

LECTURES  ON  ANCIENT  HISTORY, 

FROM  THE  EARLIEST  TIMES  TO  THE  TAKING  OF  ALEXANDRIA  BY  OCTAVIANUS, 
CONTAINING 

The  History  of  the  Asiatic  Nations,  the  Egyptians,  Crocks,  Macedonians, 
and  Carthaginians. 

BY   B.   G.    NIEBUHR. 

TRANSLATED  FROM  THE  GERMAN  EDITION  or  DR.  MARCUS  NIEBUHR, 

BY  DR.  LEONHARD  SCHMITZ,  F.  RS.E., 
With  Additions  and  Corrections  from  his  own  MSS.  notes. 

In  three  very  handsome  volumes,  crown  octavo,  extra  cloth,  containing  about 

fifteen  hundred  pages. 
From  the  Translator's  Preface. 

"  The  Lectures  on  Ancient  History  here  presented  to  the  English  public,  em- 
brace the  history  of  the  ancient  world,  with  the  exception  of  that  of  Rome, 
down  to  the  time  when  all  the  other  nations  and  states  ofclassical  antiquity  were 
absorbed  by  the  empire  of  Rome,  and  when  its  history  became,  in  point  of  fact, 
the  history  of  the  world.  Hence  the  present  course  of  Lectures,  together  with 
that  on  the  History  of  Rome,  form  a  complete  course,  embracing  the  whole  of 
ancient  history.  *  *  *  *  We  here  catch  a  glimpse,  as  it  were,  of  the  working 
of  the  great  mind  of  the  Historian,  which  imparts  to  his  narrative  a  degree  of 
freshness  and  suggestiveness  that  richly  compensate  for  a  more  calm  and  sober 
exposition.  The  extraordinary  familiarity  of  Niebuhr  with  the  literatures  of  all 
nations,  his  profound  knowledge  of  all  political  and  human  affairs,  derived  not 
only  from  books,  but  from  practical  life,  and  his  brilliant  powers  of  combina- 
tion, present  to  us  in  these  Lectures,  as  in  those  on  Roman  history,  such  an 
abundance  of  new  ideas,  startling  conceptions  and  opinions,  as  are  rarely  to  be 
met  with  in  any  other  work.  They  are  of  the  highest  importance  and  interest 
to  all  who  are  engaged  in  the  study,  uot  only  of  antiquity,  but  of  any  period  in 
the  history  of  man." 

The  value  of  this  work  as  a  book  of  reference  is  greatly  increased  by  a  very 
extensive  Index  of  about  fifty  closely  printed  pages,  prepared  by  John  Robson, 
B.  A.,  and  containing  nearly  ten  thousand  references;  in  addition  to  which 
each  volume  has  a  very  complete  Table  of  Contents. 


MEMOIRS  OF  THE  LIFE  OF  WILLIAM  WIRT. 
BY  JOHN    P.    KENNEDY. 

SECOND  EDITION,  REVISED. 

In  two  handsome  12mo.  volumes,  with  a  Portrait  and  fac-simile  of  a  letter  from 
John  Adams.     Also, 

A  HANDSOME  LIBRARY  EDITION.  IJi  TWO  BEAUTIFULLY  PRINTED  OCTAVO  VOLtTMeS. 

Ill  its  present  neat  and  convenient  form,  the  work  is  eminenlly  filled  10  assume  the 
position  which  it  nif-iiis  as  a  hook  for  every  parlor  table  and  for  every  fireside  where 
there  is  an  appreciation  of  the  kindliness  and  manliness,  the  intellect  and  the  affec- 
tion, the  wit  and  liveliness  which  rendered  William  Win  at  once  so  eminent  in  the 
world,  so  brilliant  in  society,  and  so  loving  and  loved  in  the  retiremrntof  his  domestic 
circle.  Uniting  all  these  attractions,  it  cannot  fail  to  find  a  place  in  every  private  and 
public  library,  and  in  all  collections  of  books  tor  the  use  of"  schools  and  college?;  lor 
the  young  cun  have  before  them  no  brighter  example  of  what  can  be  accomplished  by 
industry  and  resolution,  than  the  life  of  William  Win,  as  unconsciously  relaied  by 
himself  in  these  volumes. 

HISTORY  OF  THE  PROTESTANT  REFORMATION  IN  FRANCE, 
BY   MRS.   MARSH, 

Author  of  "Two  Old  Men's  Tales,"  "Emilia  Wyndham,"  &c 
In  two  handsome  volumes,  royal  12mo.,  extra  cloth. 


BLANCHARD  &  LEA'S  PUBLICATIONS.— (History  <$•  Biography.)    3 
NEW    AND    IMPROVED   EDITION. 

LIVES  OF  THE  QUEENS  OF  ENGLAND, 

FROM  THE  NORMAX  CONQ.UEST. 

WITH   ANECDOTES   OF   THEIR   COURTS. 

Now  first  published  from  Official  Records,  and  other  Authentic  Documents  Private 
as  well  as  Public. 

NEW   EDITION,   WITH   ADDITIONS   AND   CORRECTIONS. 

BY  AGNES  STRICKLAND. 

In  six  volumes,  crown  octavo,  extra  crimson  cloth,  or  half  morocco,  printed  on 

fine  paper  and  large  type. 
Copies  of  the  Duodecimo  Edition,  in  twelve  volumes,  may  still  be  hud. 

A  valuable  contribution  to  historical  knowledge,  to  young  persons  especially.  Il  con- 
tains a  mass  of  every  kind  of  historical  matter  of  interest,  which  industry  and  resource 
could  collect.  We  have  derived  much  entertainment  and  instruction  from  the  work  — 
Athtnaum. 

The  exr  cution  of  this  work  is  equal  to  the  conception  Great  pains  have  been  taken 
to  make  it  both  interesting  and  valuable. — Literary  Gazette 

A  charming  work— full  of  interest,  at  once  serious  and  pleasing.— Monsieur  Guizot. 


LIVES  OF  THE  QUEENS  OF  HENRY  VIII. 

AND  OF  HIS  MOTHER,  ELIZABETH  OF  YORK, 

BY  MISS  STRICKLAND. 
Complete  in  one  handsome  crown  octavo  volume,  extra  cloth.     (Just  Issued.) 


MEMOIRS   OF  ELIZABETH, 

SECOND  QUEEN  REGNANT  OP  ENGLAND  AND  IRELAND. 

BY  MISS  STRICKLAND. 
Complete  in  one  handsome  crown  octavo  volume,  extra  cloth.    (Just  Issued.) 


(NOW  READY.) 
MEMORIALS    AND    CORRESPONDENCE 

OF 

CHARLES    JAMBS   FOX. 
EDITED  BY  LORD  JOHN  RUSSELL. 

In  two  very  handsome  volumes,  royal  12mo.,  extra  cloth. 


THE  GARDENER'S  DICTIONARY. 

A  DICTIONARY  OF  MODERN  GARDENING.  By  G  W.  John*on,Esq.  With  nu- 
merous additions,  by  David  Landreth.  With  one  hundred  and  eighty  wood-cms. 
In  one  very  large  royal  12mo  volume,  of  about  650  double-columned  pagei. 


(NOW  READY.) 

WISE  SAWS  AND  MODERN  INSTANCES. 

BY  THE  AUTHOR  OF  SAM  SLICK. 
In  one  royal  I2mo.  volume. 


4  BLANCHARD  &  LEA'S  PUBLICATIONS.— (Miscellaneous.) 

THE  ENCYCLOPEDIA  AMERICANA^ 

A  POPULAR  DICTIONARY  OF  ARTS,  SCIENCES.  LITERATURE,  HIS- 
TORY, POLITICS,  AND  BIOGRAPHY. 

In  fourteen  large  octavo  volumes  of  over  600  double-columned  pages  each. 
For  sale  very  low,  in  various  styles  of  binding. 

Some  years  having  elapsed  since  the  original  thirteen  volumes  of  the  ENCY- 
CLOPAEDIA AMERICANA  were  published,  to  bring  it  up  to  the  present  day, 
with  the  history  of  that  period,  at  the  request  of  numerous  subscribers,  the  pub- 
lishers have  issued  a 

SUPPLEMENTARY  VOLUME  (THE  FOURTEENTH), 

BRINGING   THE    WORK   THOROUGHLY   UP. 

Edited  by  HENRY  VETHAKE,  LL.  D. 

In  one  large  octavo  volume,  of  over  650  double-columned  pages,  which  may  be 
had  separately,  to  complete  sets. 


MURRAY'S  ENCYCLOP/EDIA  OF  GEOGRAPHY. 
THE  ENCYCLOPAEDIA  OF  GEOGRAPHY,  comprising  a  Complete  Description 
of  the  Earth,  Physical,  Statistical,  Civil,  and  Political;  exhibiting  its  Rela- 
tion to  the  Heavenly  Bodies,  its  Physical  Structure,  The  Natural  History  of 
each  Country,  and  the  Industry,  Commerce,  Political  Institutions,  and  Civil 
and  Social  State  of  all  Nations.  By  HUGH  MURRAY,  F.  R.  S.  E.,  &c.  Assisted 
in  Botany,  by  Professor  Hooker — Zoology,  &c.,  by  W.  W.  Swainson — Astrono- 
my, &c.,  by  Professor  Wallace — Geology,  &c.,  by  Professor  Jameson.  Re- 
vised, with  Additions,  by  THOMAS  G.  BRADFORD.  The  whole  brought  up,  by 
a  Supplement,  to  1843.  In  three  large  octavo  volumes  various  styles  of 
binding. 

This  great  work,  furnished  at  a  remarkably  cheap  rate,  contains  about  NINKTBES 
HUNDRED  LARGE  IMPERIAL  PAGES,  and  is  illustrated  l>y  EIGHTY-TWO  SMALL  MAPS,  and  a 
colored  MAP  OF  THE  UNITED  STATES,  alter  Tanner's,  together  with  about  ELEVE.N  Hcs- 
DBKD  WOOD  CUTS  executed  iu  the  best  st>  le. 


YOU  ATT  AND  SKINNER  ON  THE  HORSE. 

THE     HORSE. 

BY    WILLIAM   YOUATT. 

./  nrtc  edition,  with  numerous  Illuttrattunt. 

TOGETHER  WITH  A  GENERAL    HISTORY   OF    THE    HORSE  J    A  DISSERTATION    ON    THE 
AMERICAN  TROTTING  HORSE  ;    HOW  TRAINED  AND  JOCKEYED  J    AN  ACCOUKT 

ASS   AND   THE    MULE. 

BY   J.    S.    SKINNER, 
Assistant  Postmaster-General,  and  Editor  of  the  Turf  Register. 

This  edition  of  Yonatfs  well-known  and  standard  work  on  the  Management,  Dis- 
eases, and  I'reaimentot  the  Horse,  embodying  the  valuable  additions  of  Mr.  Skinner,  has 
already  obtained  such  a  wide  circulation  throughout  the  country,  that  the  Publishers 
need  say  nothing  to  attract  10  it  ihe  attention  and  confidence  of  all  who  keep  Horses  or 
are  interested  in  their  improvement. 


vo 


UATT   AND    LEWIS   ON    THE    DOG. 


THE  DOG.    By  William  Youait.    Edited  l>y  E.  J  Lewis,  M.  D      VVth  numerous  and 
beautiful  illustrations.    In  one  very  handsome  volume,  crown  8vo  .crimson  clolh,  gilt. 


BLANCHARD  &  LEA'S  PUBLICATIONS.— (Science.)  5 

LIBRARY  OF  ILLUSTRATED  SCIENTIFIC  WORKS, 

A  series  of  beautifully  printed  volumes  on  various  branches  of  science,  by  the 
most  eminent  men  in  their  respective  departments.  The  whole  printed  in  the 
handsomest  style,  and  profusely  embellished  in  the  most  efficient  manner. 

fD"  No  expense  hag  been  or  will  be  spared  lo  render  this  series  worihy  of  the  support 
ot  ihe  scientific  public,  while  at  the  same,  lime  it  is  one  of  the  handsomest  specimens  of 
typographical  and  artistic  execution  which  have  appeared  in  this  country. 

UK  LA  BK CUE'S  GKOI.OGY-.Tust  Issued.) 

THE  GEOLOGICAL  OBSERVER. 

BY  SIR  HENRY  T.  DE  LA  BECHE,  C.  B.,  F.  R.  S., 

Director-General  of  the  Geological  Survey  of  Great  Britain,  kc. 

In  one  very  largo  and  handsome  octavo  volume. 
WITH    OVER    THREE    HUNDRED    WOOD-CUTS. 

We  have  here  presented  to  us,  by  one  admirably  qualified  for  the  task,  the  most  com- 
plete compendium  of  the  science  of  geology  ever  produced,  in  which  the  different  factg 
which  fall  under  the  cognizance  of  this  branch  of  natural  science  are  arranged  under 
the  different  causes  by  which  they  are  produced.  From  the  style  in  which  ibe  subject 
is  treated,  the  work  is  calculated  not  only  for  the  use  of  the  professional  geologist  but 
for  that  of  the  uninitiated  reader,  who  will  find  in  it  much  curious  and  interesting  infor- 
mation on  ihe  changes  which  the  surface  of  our  globe  has  undergone,  and  the  history  of 
the  various  striking  appearances  which  it  presents  Voluminous  as  the  work  is.  it  is 
not  rendered  unreadable  from  its  bulk,  owing  to  the  judicious  subdivision  of  its  contents, 
and  the  copious  index  which  is  appended.— John  Bull. 

Having  had  such  abundant  opportunities,  no  one  could  be  found  so  capable  of  direct- 
ing the  labors  of  the  young  geologist,  or  10  aid  by  his  own  experience  ihe  sludifs  of  those 
who  may  not  have  l>een  able  to  range  so  extensively  over  the  earth's  surface.  We 
strongly  recommend  Sir  Henry  De  la  Beche's  book  lo  those  who  desire  lo  know  what 
has  been  done,  and  to  learn  something  of  the  wide  examination  which  yet  lies  waiting 
for  the  industrious  observer.—  The  Athenaeum. 


KNAPP'S    CHEMICAL    TECHNOLOGY. 

TECHNOLOGY;  or,  CHEMISTRY  APPLIED  TO  THE  ARTS  AND  TO  MANUFACTURES. 
By  DR.  F.  KNAPP,  Professor  at  the  University  of  Giessen.  Edited,  with  nu- 
merous Notes  and  Additions,  by  DR.  EDMUND  RONALDS,  and  Da.  THOMAS 
RICHARDSON.  First  American  Edition,  with  Notes  and  Additions  by  Prof. 
WALTER  R.  JOHNSON.  In  two  handsome  octavo  volumes,  printed  and  illus- 
trated in  the  highest  style  of  art,  with  about  500  wood-engravings. 

The  style  of  excellence  in  which  the  first  volume  was  got  up  is  fully  preserved  in  lhi«. 
The  treatises  themselves  are  admirable,  and  the  editing. bolh  by  ihe  English  and  Ameri- 
can editors,  judicious;  ?o  that  ihe  work  mainiains  iifelf  as  the  best  of  the  series  to  which 
it  belongs,  and  worthy  the  attention  of  all  interested  in  the  arts  of  which  it  treats.— 
Franklin  Institute  Journal. 


WEISBACH'S_MECHANICS. 

PRINCIPLES  OF  THE  MECHANICS  OF  MACHINERY  AND  ENGINEER- 
ING.   By  PROFESSOR  JULIUS  WEISBACH.     Translated  and  Edited    by  PROF. 
GORDON,  of  Glasgow.     First  American  Edition,  with  Additions  by  PROF.  WAL- 
TER R.  JOHNSON.     In  two  octavo  volumes,  beautifully  printed,  with  90( 
trations  on  wood. 
The  most  valuable  contribution  to  practical  science  that  hag  yet  appeared  in  this 

CTi?erq^fe'lTy^a"7;ningof  ihe  kind  yet  produced  in  this  countrv-the  mo.t  standard 
book  on  mechanics,  machinery,  and  ei.jrine«ri.ii{  now  exiunt.-  N.  V.  Commercial 

In  every  way  worihy  of  being  recommended  lo  our  readers  -tranklin 
Journal. 


BLANCHARD  <fe  LEA'S  PUBLICATIONS.—  (Science.) 


8CMEJWIFIC  1MB  RA  B  »'—  (Contin  w«4.) 
CARPENTER'S    COMPARATIVE    PHTSIOLOGY-(Just  Issued.) 

PRINCIPLES  OF  GENERAL  AND  COMPARATIVE  PHYSIOLOGY;  in- 
tended as  an  Introduction  to  the  Study  of  Human  Physiology,  and  as  a  Guide 
to  the  Philosophical  Pursuit  of  Natural  History.  By  WILLIAM  B.  CARPENTER, 
M.  D.,  F.  R.  S.,  author  of  "  Human  Physiology,"  "  Vegetable  Physiology," 
&c.  &c.  Third  improved  and  enlarged  edition.  In  one  very  large  and  hand- 
some octavo  volume,  with  several  hundred  beautiful  illustrations. 


MULLER'S    PHYSICS. 

PRINCIPLES  OF  PHYSICS  AND  METEOROLOGY.  By  PROFESSOR  J.  Mui> 
LER,  M.  D.  Edited,  with  Additions,  by  R.  EGLF.SFELD  GRIFFITH,  M.  D.  In 
one  large  and  handsome  octavo  volume,  with  550  wood-cuts  and  two  colored 
plates. 

The  style  in  which  the  volume  is  published  is  in  the  highest  degree  creditable  to  the 
enterprise  of  the  publishers.  Itcoivams  near'y  four  hundred  engravings  executed  in 
a  st>  le  of  extraordinary  elegance.  We  commend  the  l>ook  to  geueral  favor.  It  is  the 
best  of  its  kind  we  have  ever  seen.— N.  Y.  Courier  and  Enquirer. 


MOHR,    REDWOOD,   AND   PROCTER'S  PHARMACY. 

PRACTICAL  PHARMACY:  Comprising  the  Arrangements,  Apparatus,  and 
Manipulations  of  the  Pharmaceutical  Shop  and  Laboratory.  By  FHANCIS 
MOHR,  Ph.  D.,  Assessor  Pharmacis  of  the  Royal  Prussian  College  of  Medicine, 
Coblentz;  and  THEOPHILUS  REDWOOD,  Professor  of  Pharmacy  in  the  Pharma- 
ceutical Society  of  Great  Britain.  Edited,  with  extensive  Additions,  by  PROF. 
WILLIAM  PROCTER,  of  the  Philadelphia  College  of  Pharmacy.  In  one  hand- 
somely printed  octavo  volume,  of  670  pages,  with  over  500  engravings  on 
wood. 


THE   MILLWRIGHT'S   GUIDE. 

THE  MILLWRIGHT'S  AND  MILLER'S  GUIDE.  By  OLIVER  EVA\S.  Eleventh  Edi- 
tion. With  Additions  and  Correction?  by  ihe  Professor  of  Mechanics  in  the  Franklin 
Institute,  and  a  description  of  an  improved  Merchant  Flour  Mi'l.  By  C.  and  O.  Evans. 
In  one  octavo  volume,  with  numerous  engravings. 

HUMAN  HEALTH  :  or.  ihe  Influence  of  Atmosphere  and  Locality.  Change  of  Air  and 
Climate,  Seasons.  Food.  Clothing.  Bathing.  Mineral  Springs.  Exercise.  Slrrp.  Corporeal 
and  Mental  Pursuits,  &c.  &c..  on  Healthy  Man.  constituting  Elements  of  Hygiene. 
By  Robley  Dunglison,  M  D.  In  one  octavo  volume. 


ACTON'S    COOKERY. 

MODERN  COOKERY  IN  ALL,  1  rS  BRANCHES,  reduced  to  a  System  of  Easy  Prac- 
tice, for  ihe  Use  of  Private  Families;  in  a  Series  of  Practical  Rt-ceipis.  all  of  which 
are  given  with  the  most  minuf  exactness.  By  Kliza  Acton  With  numerous  wood- 
cut illustralions;  to  which  i*  added  a  Table  of  Weighis  a»d  Measures.  The  whole 
revised,  and  prepared  for  American  housekeepers,  by  Mrs.  Sarah  J.  Hale.  From  the 
Second  London  Edition.  In  one  large  12mo.  volume. 

THE   DOMESTIC  MANAGEMENT  OF   THE   SICK-ROOM,  necessary,  in   aid  ot 

medical  treatment  for  the  cure  of  diseases.    By  A.  T.  Thomson,  M.  D.   Edited  by  R.  E. 

Griffith,  M.  D.    In  one  volume  royal  l'->mo  ,  extra  cloth. 
LANGUAGE   OF    FLOWERS,  with    illustrative  poetry.     Eighth  edition.    In 

one  beautiful  volume,  royal  ISmo..  crimson  cloth,  gilt,  with  colored  plates. 
AMERICAN  ORNITHOLOGY".    By  Charles  Bonaparte.Prince  of  Canino.  In  four  folio 

volumes,  half  bound,  wnh  numerous  magnificent  colored  plates. 

LECTURES  ON  THE  PHYSICAL  PHENOMENA  OF  LIVING  BEINGS.  By 
Carlo  Maiteupci.  Edited  by  Jonathan  I'ereira,  M  D.  In  one  royal  12mo.  volume, 
extra  cloth,  wilh  illustrations. 


BLANCHARD  &  LEA'S  PUBLICATIONS.— (Science.)  7 

GRAHAMS  CHEMISTRY,  NEW  EDITION.    Part  I.-lNow  Ready.) 

ELEMENTS  OF  CHEMISTRY; 

INCLUDING   THE   APPLICATIONS   OF   THE  SCIENCE  IN  THE  ARTS. 

BY  THOMAS   GRAHAM,  F.  R.  S.,  &.C., 

Professor  of  Chemistry  in  University  College,  Ixtndon,  Ac. 

Second  American,  from  an  entirely  Revised  and  greatly  Enlarged  English  Edition. 

WITH  NUMEROUS  WOOD-ENGRAVINGS, 

EDITED,  \VITH  NOTES,  BY  ROBERT  BRIDGES.  M.  D., 
Professor  of  Chemistry  in  the  Philadelphia  College  of  Pharmacy,  &c. 

To  be  completed  in  Two  Parts,  forming  one  very  large  octavo  volume. 
PART  I,  now  ready,  of  430  large  pages,  with  185  engravings. 
PART  II,  preparing  for  early  publication. 

From  the  Editor's  Preface. 

The  "  Elements  of  Chemistry,"  of  which  a  second  edition  is  now  presented, 
attained,  on  its  first  appearance,  an  immediate  and  deserved  reputation.  The 
copious  selection  of  facts  from  all  reliable  sources,  and  their  judicious  arrange- 
ment, render  it  a  safe  guide  for  the  beginner,  while  the  clear  exposition  of  the- 
oretical points,  and  frequent  references  to  special  treatises,  make  it  a  valuable 
assistant  for  the  more  advanced  student. 

From  this  high  character  the  present  edition  will  in  no  way  detract.  The 
great  changes  which  the  science  of  Chemistry  has  undergone  during  the  interval 
have  rendered  necessary  a  complete  revision  of  the  work,  and  this  has  been 
most  thoroughly  accomplished  by  the  author.  Many  portions  will  therefore  be 
found  essentially  altered,  thereby  increasing  greatly  the  size  of  the  work,  while 
the  series  of  illustrations  has  been  entirely  changed  in  style,  and  nearly  doubled 
in  number. 

Under  these  circumstances  but  little  has  been  left  for  the  editor.  Owing, 
however,  to  the  appearance  of  the  London  edition  in  parts,  some  years  have 
elapsed  since  the  first  portions  were  published,  and  he  has  therefore  found  oc- 
casion to  introduce  the  more  recent  investigations  and  discoveries  in  some  sub- 
jects, as  well  as  to  correct  such  inaccuracies  or  misprints  as  had  escaped  the 
author's  attention,  and  to  make  a  few  additional  references. 

INTRODUCTION  TO  PRACTICAL  CHEMISTRY,  including  Analysis.  By 
J«hn  E.  Bowman,  M.  D.  In  one  neat  royal  12mo.  volume,  extra  cloth,  with  numer- 
ous illustrations. 

DANA   ON    CORALS. 
ZOOPHYTES  AND  CORALS.     By  James  D.  Dana.    In  one  volume  imperial 

qiiano.  extra  cloth,  wilh  wood-cuts. 

Also,  an  Atlas  to  the  above,  one  volume  imperial  folio,  wilh  sixty-one  magnificent 
pla:es.  colored  after  nature.  Bound  in  half  morocco. 

Tht-se  splendid  volumes  form  a  portion  of  the  publication?  of  the  United  States  Explor- 
ing Expedition.  As  but  very  few  copies  have  been  prepared  for  sale,  and  as  lhes« 
are  nearly  exhausted,  all  who  are  desirousof  enrichingtheir  libruri.s  with  this,  the  most 
creditable  specimen  of  American  Art  and  Science  as  yet  issued,  will  do  well  lo  procure 
copies  at  once. 

THE  ETHNOGRAPHY  AND  PHILOLOGY  OF  THE  UNITED  STATES  EX- 
PLORING EXPEDITION.  By  Horatio  Hale.  In  one  large  imperial  quarto  volume, 
beautifully  printed,  and  strongly  bound  in  extra  cloth. 

BARON   HUMBOLDT'S    LAST    WORK. 
ASPECTS    OF    NATURE    IN    DIFFERENT    LANDS    AND    DIFFERENT 

CLIMATES  With  Scieniific  Elucidations.  By  Alexander  Von  Humboldt.  Trans- 
lated by  Mrs.  Sabine.  Second  American  edition.  In  one  handsome  volume,  large 
royal  l'2mo.,  extra  cloth- 

CHEMISTRY  OF  THE  FOUR  SEASONS,  SPRING,  SUMMER,  AUTUMN,  AND 
WINTER  By  Thomas  Griffiih.  In  one  handsome  volume,  royal  I2rno ,  extra  cloth, 
with  numerous  illustrations. 


8     BLANCHARD  &  LEA'S  PUBLICATIONS.— (Educational  Works.) 
A  NEW  TEXT-BOOK  ON  NATURAL  PHILOSOPHY. 

HANDBOOKS 

OF  NATURAL  PHILOSOPHY  AND  ASTRONOMY. 

BY  DIONYSIUS  LARDNER,  LL.  D.,  ETC. 

FIRST  COURSE,  containing 

Mechanics,  Hydrostatics,  Hydraulics,  Pneumatics,  Sound,  and  Optics, 

la  one  large  royal   12mo.  volume  of  750  pages,  strongly  bound  in  leather,  with 
over  400  wood-cuts,  (Just  Issued.) 

THE  SECOND  COURSE,  embracing 

HEAT,  MAGNETISM,  ELECTRICITY,  AND  GALVANISM, 

Of  about  400  pages,  and  illustrated  with  250  cuts,  is  now  ready. 

THE  THIRD  COURSE,  constituting 
A   COMPLETE    TREATISE   ON    ASTRONOMY 

WITH    NUMEROUS   STEEL   PLATES   AND   WOOD-CUTS,    IS   NEARLY   READY. 

The  intention  of  the  author  has  been  lo  prepare  a  work  which  should  embrace  the 
principles  of  Natural  Philosophy,  in  iheir  laiesi  state  of  scientific  development,  divested 
of  Ihe  sbslruseness  which  renders  them  unfitted  for  the  younger  student,  and  at  the  same 
time  illustrated  by  numerous  practical  applications  111  every  branch  of  art  and  science. 
Dr.  Lardner's  extensive  acquirements  in  all  departments  of  human  knowledge,  and  his 
well-known  skill  in  popularizing  his  subject,  have  thus  enabled  him  to  present  a  text- 
book which,  though  strictly  scientific  in  its  groundwork,  is  yet  easily  mastered  by  the 
student,  while  calculated  to  interest  ihe  mind,  and  awaken  the  attention  by  showing  the 
importance  of  the  principles  discussed,  and  the  manner  in  which  they  maybe  made 
subservient  lo  the  practical  purposes  of  life  To  accomplish  this  Mill  furtrte r.  the  editor 
has  added  to  each  section  a  serie*  of  examples,  to  be  worked  out  by  the  learner,  thus 
impressing  upon  him  the  practical  importance  and  variety  of  the  results  to  be  obtained 
from  the  general  laws  of  nature.  The  subject  is  siill  further  simplified  by  the  very  large 
number  of  illustrative  wood-cuts  which  are  scattered  through  the  volume,  making  plain 
to  the  eye  what  might  not  rea.lily  be  grasped  by  the  unassisted  mind  ;  and  every  care 
has  been  taken  to  render  the  typographical  accuracy  of  the  work  what  it  should  be. 

Although  the  first  portion  only  IIHS  been  issued,  and  that  but  for  a  few  months,  yet  il 
has  already  been  adopted  by  many  academies  and  colleges  of  the  highest  standing  and 
character.  A  few  of  the  numerous  recommendations  with  which  the  work  has  been 
favored  are  subjoined. 

From  Prof.  Millington,  Univ.  of  Mississippi,  April  10, 1852. 

I  am  highly  pleased  with  its  contents  and  arrangement.  It  contains  a  greater  number 
of  every  day  useful  practical  fads  and  examples  than  I  have  ever  seen  noticed  in  a 
similar  work,  mid  I  do  not  hesitate  to  say  that  as  a  book  for  teaching  I  prefer  it  to  any 
other  of  the  same  size  and  extent  that  I  am  acquainted  with.  During  ihe  thirteen  years' 
that  I  was  at  William  and  Mary  College  I  had  to  leach  Natural  Philosophy ,  and  I  should 
have  been  very  gladio  have  such  a  text-book. 

From  Edmund  Smith,  Baltimore,  May  19.  1S52. 

I  have  a  class  using  it,  and  think  it  the  best  book  of  the  kind  with  which  I  am  ac- 
quainted. 

From  Prof.  Cleveland,  Philadelphia,  October  17, 1851. 

I  feel  prepared  to  say  that  it  is  the  fullest  and  most  valuable  manual  upon  the  subject 
that  ha*  fallen  under  my  notice,  and  1  intend  lo  make  il  the  text  book  for  the  first  class 
in  my  school. 

From  S.  Schooler,  Hanorer  Academy,  Fa.. 

The  "  Handbooks"  seem  tome  the  best  popular  treatises  on  their  respective  subjects 
whh  which  I  am  acquainted.  Dr.  Lardner  certainly  popularizes  science  very  well,  and 
a  good  lexi-book  for  schools  and  colleges  was  noi  before  in  exisience. 

From  Prtf.  J.  S.  Henderson,  Farmer's  College,  O,  Feb.  16, 1S52. 

It  is  an  admirable  work,  and  well  worthy  of  publie  patronage.  For  clearness  and 
fulness  il  is  unequalled  by  any  thai  1  have  seen. 


BLANCHAKD  &  LEA'S  PlTBLICATIONS.-(£^««zi!t0«0/  Works.}     9 
NEW  AND  IMPROVED  EDITION.— (Now  Ready.) 

OUTLINES    OF    ASTRONOMY. 

BY  SIR  JOHN  F.  W.  HERSCIIEL,  F.  R.  S.,  &c. 

A  MEW  AMERICAN  FROM  THE  FOURTH' LONDON  EDITION. 

In  one  very  neat  crown  octavo  volume,  extra  cloth,  with  six  plates  and  nu- 
merous wood-cuts. 

This  edition  will  be  found  thoroughly  brought  up  to  the  present  state  of  as- 
tronomical science,  with  the  most  recent  investigations  and  discoveries  fully 
discussed  and  explained. 

We  now  take  leave  of  this  remarkable  work,  which  we  hold  to  he.  beyond  a  doubt, 
the  grealest  and  most  remarkable  of  the  works  in  which  the  laws  of  astronomy  and  the 
appearance  of  the  heavens  are  described  10  those  who  are  not  mathematicians  nor  ob- 
servers, and  recalled  to  those  who  are.  It  is  the  reward  of  men  who  can  descend  from 
the  advancement  of  knowledge  to  care  for  its  diffusion,  that  their  works  are  essential 
to  all.  that  they  become  the  manuals  of  the  proficient  as  well  as  the  text-books  of  the 
learner.—  Athtnrr.um. 

There  is  perhaps  no  book  in  the  English  language  on  the  subject,  which,  whilst  it  con- 
tains so  many  of  the  facts  of  Astronomy  (which  it  attempt*  to  explain  with  as  little  tech- 
nical language  as  possible),  is  so  attractive  in  its  style,  and  so  clear  and  forcible  in  its 
illustrations.—  Evangelical  Review. 

Probably  no  book  ever  written  upon  any  science,  embraces  within  so  small  a  compass 
an  entire  epitome  of  everything  known  within  all  its  various  departments,  practical, 
theoretical,  and  physical.—  Examiner. 


A  TREATISE  ON,  ASTRONOMY. 
BY  SIR  JOHN  F.  W.  HERSCHEL.    Edited  by  S.C.  WALKER.    In  one  12mo. 

volume,  half  bound,  with  plates  and  wood-cuts. 


A    TREATISE    ON    OPTICS. 

BY  SIR  DAVID  BREWSTER,  LL.  D.,  F.  R.  S.,  &o. 
A  NEW  EDITION. 

WITH  AN  APPENDIX,  CONTAINING    AN    ELEMENTARY    VIEW    OF    THE    APPLICATION 

BY  A.  D.  BACHE,  Superintendent  U.  S.  Coast  Survey,  &c. 
In  one  neat  duodecimo  volume,  half  bound,  with  about  200  illustrations. 


BOLMAR'S  FRENCH  SERIES. 

New  editions  ofthe  following  works,  by  A.  BOLMAH,  forming,  in  connection 
with  "  Bolmar's  Levizac,"  a  complete  series  for  the  acquisition  of  the  French 
language  : — 
A.  SELECTION  OF  ONE  HUNDRED   PERRIN'S  FABLES,  accompanied  hy 

to  (wTn^ouune'diffe'reiice  between  the  French  and  English  idiom.  &c.  In  one  vol  12mo. 

A  COLLECTION  OF  COLLOQUIAL  PHRASES,  on  every  topic  necessary  to 
maintain  conversation.  Arranged  under  different  heads,  with  numerous  remark*  on 
the  peculiar  pronunciation  and  uses  of  various  words  ;  the  whole  so  deposed  a*  con- 
siderably to  facilitate  the  acquisition  of  a  correct  pronunciation  of  the  French.  In 
one  vol.  18mo 

LES  AVENTURES  DE  TELEMAQUE,  PAR  FENELON,  in  one  vol.  12roo., 
accompanied  by  a  Key  to  the  tirst  eight  hooks.  In  one  vol.  ISmo.,  containing,  like  the 
Fables,  the  Text,  a  literal  and  free  translation,  intended  as  a  sequel  to  the  Fables. 
Either  volume  sold  separately. 

ALL  THE  FRENCH  VERBS,  both  regular  and  irregular,  in  a  small  volume. 


10    BLANCHAKD  &  LEA'S  PUBLICATIONS.— (Educational  Wor&s.) 

ELEMENTS  OF  NATURAL  PHILOSOPHY; 

BEING 

AN  EXPERIMENTAL  INTRODUCTION  TO  THE  PHYSICAL  SCIENCES. 
Illustrated  with  over  Three  Hundred  Wood-cuts. 

BY   GOLDING  BIRD,   M.D., 

Assistant  Physician  to  Guy's  Hospital. 
From  the  Third  London  edition.     In  one  neat  volume,  royal  12tno. 

We  are.  astonished  to  find  that  there  is  room  in  so  email  a  book  for  even  the  hare 
recital  of  so  many  subject?.  Where  everything  is  treafd  succinctly,  great  judgment 
and  much  time  are  needed  in  making  a  selection  and  winnowing  the  wheat  from  the 
chaff  Dr.  Bird  has  no  need  to  plead  the  peculiarity  of  his  position  as  a  shield  against 
criticism,  so  Ion?  as  his  hook  continues  to  be  the  best  epitome  in  the  English  lan- 
guage of  this  wide  range  of  physical  subjects. — North  American  Review,  April  1,  1851. 

From  Prof  John  Johnston,  Wesleyan  Univ.,  Middletown.  Ct. 

For  those  desiring  as  extensive  a  work,  I  think  it  decidedly  superior  to  anything  of 
the  kind  with  which  I  am  acquainted. 

From  Prof.  R.  O.  Carrey,  East  Tennessee  University. 

I  am  much  gra'ified  in  perusing  a  work  which  so  well,  so  fully,  and  so  clearly  sets 
forth  i his  branch  of  the  Natural  Sciences.  For  some  time  I  have  been  desirous  of  ob- 
taining a  substitute  for  the  one  now  used— one  which  should  embrace  the  recent  dis- 
coveries in  the  sciences,  and  I  can  truly  say  that  such  a  one  is  afforded  in  this  work  of 
Dr.  Bird's. 

From  Prof.  W.  F.  Hopkins,  Masonic  University,  Tenn. 

It  is  just  the  sort  of  book  I  think  needed  in  most  colleges,  being  far  above  the  rank  of 
a  mere  popular  work,  and  yet  uot  beyond  the  comprehens;on  of  all  but  the  most  accom- 
plished mathematicians.  •• 


ELEMENTARY  CHEMISTRY; 

THEORETICAL   AND    PRACTICAL. 
BY  GEORGE  FOWXES,  PH.  D., 

Chemical  Lecturer  in  the  Middlesex  Hospital  Medical  School,  &c.  &c. 

WITH  NUMEROUS  ILLUSTRATIONS. 
Third  American,  from  a  late  London  edition.     Edited,  with  Additions, 

BY  ROBERT  BRIDGES,  M.  D., 

Professor  of  General  and  Pharmaceutical  Chemistry  in  the  Philadelphia 

College  of  Pharmacy,  &c.  Sec. 

In  one  large  royal  12mo.  volume,  of  over  five  hundred  pages,  with  about  180 
wood-cuts,  sheep  or  extra  cloth. 

The  work  of  Dr.  Fownes  ha*  long  been  before  the  public,  and  its  merits  have  been 
fully  appreciated  as  the  best  text  book  on  Chemistry  now  in  exigence.  We  do  not,  of 
course,  place  it  in  a  rank  superior  to  the  works  of  Brande.  Graham.  Turner.  Gregory, 
or  Gmelin.  hut  we  say  that,  as  a  work  for  students,  it  is  preferable  to  any  of  them.— Lou- 
don  Journal  of  Medicine. 

We  kno*  of  no  treatise  eo  well  calculated  to  aid  the  student  in  becoming  familiar 

a  text  book  for  tiiose  attending  Chemical  Lecture*.  *  *  *  *  The  best  text-book  on  Che- 
mistry that  has  issued  from  our  press. — American  Med.  Journal. 

We  know  of  none  within  the  same  limits,  which  has  higher  claims  to  our  confidence 
as  H  collect;  class-book,  both  for  accuracy  of  detail  and  scientific  arrangement. — Au- 
gusta Med  Journal. 


ELEMENTS    OP    PHYSICS. 

OR.  NATURAL  PHILOSOPHY.  GENERAL  AND  MEDICAL  Written  tor  uni- 
versal u«e.  in  plain,  or  non-technical  language  By  NEILL  AR.NOTT,  M.  D.  In  one 
octavo  volume,  with  about  two  hundred  illustrations. 


BLAXCHARD  &  LEA'S  PUBLICATIONS.-^****™*/  Worts.)    11 
NEW  AND  IMPROVED  EDITION.—  (Now  Ready.) 

PHYSICAL    GEOGRAPHY. 

BY  MARY   SOMERVILLE. 

A    NEW    AMERICAN    FROM    THE    LAST    AND    REVISED    LONDON    EDITION. 
WITH  AMERICAN  NOTES,  GLOSSARY,  ETC. 

BY  W.  S   W.  RUSCHENBERGER,  M.  D.,  U.  S.  N. 

In  one  neat  royal  12mo.  volume,  extra  cloth,  of  over  five  hundred  and  fifty  pages. 

The  great  success  of  this  work,  and  its  introduction  imo  m;iny  of  our  higher  schools 
and  academies,  have  induced  the  publishers  10  prepare  a  new  and  much  improved 
edition.  In  addition  to  the  corrections  and  improvements  of  the  author  bestowed  on 
the  work  in  its  passage  through  the  press  a  setond  time  in  London,  notes  have  been 
introduced  10  adapt  u  more  fully  lo  the  physical  geography  of  this  country;'  and  a 
comprehensive  glossary  has  been  added,  rendering  the  volume  more  particularly  suited 
to  educational  purposes. 

Our  praise  comes  lagging  in  the  rear,  and  is  wellnigh  superfluous.  But  we  are 
anxious  to  recommend  to  our  youth  the  enlarged  method  of  »  inlying  geography  which 
her  present  work  demonstrates  to  be  as  captivating  as  it  is  instructive.  We  hold 
such  presents  ns  Mrs  !*omerville  has  bestowed  npon  the  public,  to  be  of  incalculable 
value,  di-^eminating  more  sound  information  than  all  the  literary  and  scientific  insti- 
tutions will  accomplish  in  a  whole  cycle  oftheir  existence.—  Ulaekitooif*  Magazine. 
From  Lifvtenanl  Maury.  U  S.  N. 

Na'ional  Observatory,  Washington.  Jvnert  1853. 

I  thank  you  for  the  "  Physical  Geography  ;"  it  is  capital.  1  have  been  reading  it.  and 
like  it  so  much  thai  I  have  made  it  a  school-book  lor  my  children,  whom  i  am  leaching. 
There  is  in  my  opinion,  no  work  upon  that  interesting  subjecion  which  ft  treats—  Physical 
Geography—  thHt  would  make  a  better  text-book  in  our  schools  anil  colleges.  }  hope  il 
will  be  aiiopled  as  such  generally,  for  youhave  Americanized  u  and  improved  it  in  oiher 


Dr.  \V.  S.  W.  RCSCHESBEEGKR,  U.  S-  N., 
Philadelphia. 

from  Thomas  Shtrwln,  High  School,  Boston. 

I  hold  it  in  the  highest  estimation,  and  am  conn  lent  that  it  will  prove  a  very  efficient 
aid  in  the  education  of  the  young,  and  a  source  of  much  interest  and  instiuciion  to  the 
adult  reader. 

From  Erastus  Everett.  High,  School,  New  Orleans. 

I  have  examined  il  with  a  good  deal  of  care,  and  am  a  lad  to  find  thai  it  supplies  an  im- 
portant desideratum.  The  whole  work  is  a  masterpiece.  Whether  we  examine  the 
importance  of  the  subjects  treated,  or  the  elegai.t  and  attractive  style  m  which  they  are 
presented  this  work  leaves  nothing  to  desire.  I  have  introduced  ,t  into  my  school  for. 
the  use  of  an  advanced  class  in  geography.  and  they  are  greatly  interested  in  IL  I  have 
no  doubt  that  u  will  be  used  in  most  of  our  higher  seminaries. 


JOHNSTON'S  PHYSICAL  ATLAS. 

THE    PHYSICAL    ATLAS 

OF  NATURAL  PHENOMENA. 

FOR  THE  USE  OF  COLLEGES,  ACADEMIES,  AND  FAMILIES. 

BY  ALEXANDER  KEITH  JOHNSTON,  F.  R.  G.  S.,  F.  G.  S. 

In  one  larve  volume,  imperial  quarto,  handsomely  and  strongly  bound.     With 

twenty-six  plates,  engraved  aud  colored  in  the  best  style.     Together 

with  one  hundred  and  twelve  pagesof  Descriptive  Letter-press, 

and  a  very  copious  Index. 
A  wo  k  which  should  be  in  every  family  and  every  school-room,  for  wn»u}iMJon  .»d 


easy  ot  reference 


12    BLANCHARD  &  LEA'S  PUBLICATIONS.— (Educational  Works.-) 

SCHMITZ  AND  ZUMPT'S  CLASSICAL  SERIES. 

Under  this  title  BLANCHARD  &  LEA  are  publishing  a  series  of  Latin  School- 
Books,  edited  by  those  distinguished  scholars  and  critics,  Leonhard-  Schmitz 
and  C.  G.  Zutnpt.  The  object  of  the  series  is  to  present  a  course  of  accurate 
texts,  revised  in  accordance  with  the  latest  investigations  and  MSS.,  and  the 
most  approved  principles  of  modern  criticism,  as  well  as  the  necessary  element- 
ary books,  arranged  on  the  best  system  of  modern  instruction.  The  former  are 
accompanied  with  notes  and  illustrations  introduced  sparingly,  avoiding  on  the 
one  hand  the  error  of  overburdening  the  work  with  commentary,  and  on  the  other 
that  of  leaving  the  student  entirely  to  his  own  resources.  The  main  object  has 
been  to  awaken  the  scholar's  mind  to  a  sense  of  the  beauties  and  peculiarities 
of  his  author,  to  assist  him  where  assistance  is  necessary,  and  to  lead  him  to 
think  and  to  investigate  for  himself.  For  this  purpose  maps  and  other  en- 
gravings are  given  wherever  useful,  and  each  author  is  accompanied  with  a 
biographical  and  critical  sketch.  The  form  in  which  the  volumes  are  printed 
is  neat  and  convenient,  while  it  admits  of  their  being  sold  at  prices  unpre- 
cedentedly  low,  thus  placing  them  within  the  reach  of  many  to  whom  the  cost 
of  classical  works  has  hitherto  proved  a  bar  to  this  department  of  education; 
while  the  whole  series  being  arranged  on  one  definite  and  uniform  plan,  enables 
the  teacher  to  carry  forward  his  student  from  the  rudiments  of  the  language 
without  the  annoyance  and  interruption  caused  by  the  necessity  of  using  text- 
books founded  on  varying  and  conflicting  systems  of  study. 

CLASSICAL   TEXTS    PUBLISHED    INT   THIS   SERIES. 

I.  CJESARIS  DE  BELLO  GALLICO  LIBRI    IV.,  1   vol.   royal   ISmo.,  extra 

cloth,  232  pages,  with  a  Map,  price  50  cents. 

II.  C.C.  SALLUSTII  CATILINA  ET  JUGURTHA,  1  vol.  royal  ISmo.,  extra 

cloth,  168  pages,  with  a  Map,  price  50  cents. 

III.  P.  OVIDII  NASONIS  CARMINA   SELECTA,  1  vol.  royal   18mo.,  extra 
cloth,  246  pages,  price  60  cents. 

IV.  P.  VIRGILII  MARONIS  CARMINA,  1  vol.  royal  18mo.,  extra  cloth,  438 
pages,  price  75  cents. 

V.  Q.  HORATII  FLACCI  CARMINA  EXCERPTA,  1  vol.  royal  ISmo.,  extra 

cloth,  312  pages,  price  60  cents. 

VI.  Q.  CURTII   RUFl    DE    ALEXANDRI   MAGNI  QU^E    SUPERSUNT,   1 
vol.  royal  ISmo.,  extra  cloth,  326  pages,  with  a  Map,  price  70  cents. 

VII.  T.  LIVII  PATAVINI  HISTORIARUM   LIBRI  I.,  II.,  XXL,  XXII.,  1 

vol.  royal  ISuio.,  ex.  cloth,  350  pages,  with  two  colored  Maps,  price  70  cents. 

VIII.  M.  T.  CICERONIS  ORATIONES  SELECTEE  XII.,  1  vol.  royal  ISmo., 
extra  cloth,  300  pages,  price  60  cents. 

IX.  CORNELIUS  NEPOS,  I  vol.  royal  ISmo.,  price  50  cents. 
ELEMENTARY  WORKS   PUBLISHED  IN  THIS  SERIES. 

I. 

A  SCHOOL  DICTIONARY  OF   THE   LATIN   LANGUAGE.     By  DR.  J.  H. 

KALTBCHMIDT.     In  two  parts,  Latin-English  and  English-Latin. 
Part  I.,  Latin-English,  of  nearly  500  pages,  strongly  bound,  price  90  cents. 
Part  II.,  English-Latin,  of  about  400  pages,  price  75  cents. 

Or  the  whole  complete  in  one  very  thick  royal   ISmo.  volume,  of  nearly  900 

closely  printed  double-columned  pages,  strongly  bound  in  leather, 

price  only  $1   25. 

II. 

GRAMMAR  OF  THE  LATIN  LANGUAGE.  BY  LEONHARD  SCHMFTZ,  Ph. 
D.,  F.  R.  S.  E.,  Rector  of  the  High  School,  Edinburgh,  &c.  In  one  hand- 
some volume,  royal  ISmo.,  of  318  pages,  neatly  half  bound,  price  60  cents. 


BLANCHARD  &  LEA'S  PUBLICATIONS.— (Educational  Works.-)    13 
SCHM1TZ  AND  ZUMPT'S  CLASSICAL  SERIES— Continued. 


ELEMENTARY  GRAMMAR  AND  EXERCISES.  BY  DR.  LEONHAKD  SCHMITZ, 
F.  R.  S.  E.,  Rector  of  the  High  School,  Edinburgh,  &c.  In  one  handsome 
royal  18mo.  volume  of  246  pages,  extra  cloth,  price  50  cents.  (Just  Issued.) 

PREPARING    FOR    SPEEDY    PUBLICATION. 
LATIN  READING  AND  EXERCISE  BOOK,  1  vol.,  royal  )8mo. 
A  SCHOOL  CLASSICAL  DICTIONARY,  1  vol.,  royal  I8mo. 

It  will  thus  be  seen  that  this  series  is  now  very  nearly  complete,  embracing 
eight  prominent  Latin  authors,  and  requiring  but  two  more  elementary  works 
to  render  it  sufficient  in  itself  for  a  thorough  course  of  study,  and  these  latter 
are  now  preparing  for  early  publication.  During  the  successive  appearance  of 
the  volumes,  the  plan  and  execution  of  the  whole  have  been  received  with 
marked  approbation,  and  the  fact  that  it  supplies  a  want  not  hitherto  provided 
for,  is  evinced  by  the  adoption  of  these  works  in  a  very  large  number  of  the 
best  academies  and  seminaries  throughout  the  country. 

But  we  cannot  forbear   commending  esperially  both  to  instructors  and  pupils  the 

Here  will  be,  found'a  set  of  text-books  that  combine  the  excellencies  so  long  desired 
in  this  clii*?  of  works.  They  will  not  cost  ihe  student,  by  one  half  at  least,  ihm  which 
he  must  expend  for  some  other  editions.  And  who  will  not  say  that  this  is  a  consider- 
ation worthy  of  attention?  For  the  cheaper  our  school-books  can  be  made,  the  more 
widely  will  they  be  circulated  and  used.  Here  you  will  find,  too.  no  useless  display  of 
notes  iiml  of  learning,  liui  in  foot  notes  on  each  page  you  have  everything  necessary  to 
the  understanding  ol  ihe  lext.  The  diihcult  points  are  sometimes  elucidated,  and  often 
is  the  student  relerred  to  the  places  where  he  can  find  light,  but  not  without  some  effort 
of  his  own.  We  think  that  the  punctuation  in  these  books  might  be  improved;  but 
t*ken  as  a  whole,  they  come  nearer  to  the  wants  of  the  times  than  any  within  our  know- 
ledge.— Southern  College  Review. 


Uniform  with  SCHMITZ  AND  ZUMPT'S  CLASSICAL  SERIES-(Now  Ready,) 
THE   CLASSICAL  MANUAL; 

AN  EPITOME  OF  ANCIENT  GEOGRAPHY,  GREEK  AND  ROMAN 

MYTHOLOGY,  ANTIQUITIES,  AND  CHRONOLOGY. 

CHIEFLY    INTENDED    FOR   THE    USE    OF    SCHOOLS. 

BY  JAMES  S.  S.  BAIRD,  T.  C.  D., 

Assistant  Classical  Master,  King's  School,  Gloucester. 
In  one  neat  volume,  royal  ISmo.,  extra  cloth,  price  Fifty  cents. 

This  little  volume  has  been  prepared  to  meet  the  recognized  want  ofan  Epi- 
tome which,  within  the  compass  of  a  single  small  volume,  should  contain  the 
information  requisite  to  elucidate  the  Greek  and  Roman  authors  most  com- 
monly read  in  our  schools.  The  aim  of  the  author  has  been  to  embody  in  it 
such  details  as  are  important  or  necessary  for  the  junior  student,  in  a  form  and 
space  capable  of  rendering  them  easily  mastered  and  retained,  and  he  has  con- 
sequently not  incumbered  it  with  a  mass  of  learning  winch,  though  highly 
valuable" to  the  advanced  student,  is  merely  perplexing  to  the  beginner.  In  the 
amount  of  information  presented,  and  the  manner  in  which  it  is  convoyed,  as 
well  as  its  convenient  size  and  exceedingly  low  price,  it  is  therefore  adrmrably 
adapted  for  the  younger  classes  of  our  numerous  classical  schools. 


14    BLANCHARD  &  LEA'S  PUBLICATIONS.— (Educational  Works.) 
SCRIPTURE  GEOGRAPHY  AND  HISTORY.— (Just  Ready.) 
OUTLINES  OF 

SCRIPTURE  GEOGRAPHY  AND  HISTORY. 

Illustrating  the  Historical  Portions  of  the  Old  and  New  Testaments, 

DESIGNED    FOR  THE 

USE  OF  SCHOOLS  AND  PRIVATE  READING. 
BY  EDWARD  HUGHES,  F.R.A.S.,  F.  G.  S.,  &c.  &c. 

In  one  neat  volume,  royal  12mo.,  extra  cloth,  of  about  four  hundred  pages,  with 

twelve  colored  Maps. 

The  intimate  connection  of  Sacred  History  with  the  geography  and  physical 
features  of  the  various  lands  occupied  by  the  Israelites  renders  a  work  like  the 
present  an  almost  necessary  companion  to  all  who  desire  to  read  the  Scriptures 
understandingly.  To  the  young  especially,  a  clear  and  connected  narrative  of 
the  events  recorded  in  the  Bible,  is  exceedingly  desirable,  particularly  when  il- 
lustrated, as  in  the  present  volume,  with  succinct  but  copious  accounts  of  the 
neighboring  nations  and  of  the  topography  and  political  divisions  of  the  countries 
mentioned,  coupled  with  the  results  of  the  latest  investigations,  by  which 
Messrs.  Layard,  Lynch,  Olin,  Durbin,  Wilson,  Stephens,  and  others  have  suc- 
ceeded in  throwing  light  on  so  many  obscure  portions  of  the  Scriptures,  verify- 
ing their  accuracy  in  minute  particulars.  Few  more  interesting  class-books  could 
therefore  be  found  for  schools  where  the  Bible  forms  a  part  of  education,  and 
none,  perhaps,  more  likely  to  prove  of  permanent  benefit  lo  the  scholar.  The 
influence  which  the  physical  geography,  climate,  and  productions  of  Palestine 
had  upon  the  Jewish  people  will  be  found  fully  set  forth,  while  the  numerous 
maps  present  the  various  regions  connected  with  the  subject  at  their  most  pro- 
minent periods. 


HISTORY  OF  CLASSICAL  LITERATURE.— (Now  Ready.) 

HISTORY  OF  ROMAN  CLASSICAL  LITERATURE. 

BY  R.  \V.  BROWNE,  M.  A., 

Professor  of  Classical   Literature  in  King's  College.  London. 
In  one  handsome  crown  octavo  volume,  extra  cloth. 

ALSO, 
Lately  Issued,  by  the  same  author,  to  match. 

A  HISTORY  OF  GREEK  CLASSICAL  LITERATURE. 

From  Prof.  J.  A.  Spencer,  New  York,  March  19,  1852. 

It  is  an  admirable  volume,  sufficiently  full  and  copious  in  detail,  clear  and  precise  in 
style,  very  scholar-like  in  its  execution,  genial  in  its  criticism,  and  altogether  display- 
ing a  mind  well  stored  with  the  learning  genius,  wisdom,  and  exquisite  tasie  of  the 
ancient  Greeks.  It  is  in  advance  of  everything  we  have,  and  it  may  be  considered 
indispensable  to  the  classical  scholar  and  student. 

From  Prof.  N.  H.  Griffin,WilHams  College,  Mass.,  March  22.  1852. 
A  valuab'e  compend,  embracing  in  a  small  compass  matter  which  the  student  would 
have  to  go  over  much  ground  to  gather  for  himself. 


GEOGRAPHIA    CLASSICA: 

OR,  THE  APPLICATiOV  OF  ANCIENT  GEOGRAPHY  TO  THE  CLASSICS. 
By  SAMUEL  BUTLER.  D  D.,  late  Lord  Bishop  of  Litchfield.  Revised  by  his  Son.  Sixth 
American,  from  the  last  London  Edition,  with  Questions  on  the  Map*, by  JOHN  FROST, 
LL.  D.  la  one  neat  volume,  ro>al  12mo.,  half  bound. 


AN    ATLAS    OF   ANCIENT    GEOGRAPHY. 

By  SAMUEL  BUTLKR,  D.  D..  late  Lord  Bishop  of  Litchfield     In  one  octavo  volume,  half 
bound,  containing  twenty-one  quarto  colored  Maps,  and  an  accentuated  Index. 


BLANCHARD  &  LEA'S  PUBLICATIONS.-^^,,,,,/  Worjtg)    15 
ELBMENTSOirTHE~NATURAirsCm^CES-(Now  Ready.) 

THE   BOOK   OF   NATURE; 

AN  ELEMENTARY  INTRODUCTION 

TO  THE  SCIENCES  OF 

PHYSICS,  ASTRONOMY,  CHEMISTRY,  MINERALOGY,  GEOLOGY,  BOTANY, 

PHYSIOLOGY,  AND  ZOOLOGY. 
BY  FREDERICK  SCHOEDLER,  PH.  D. 

Professor  of  ihe  Natural  Sciences  at  Worms 

First  American  Edition,  with  a  Glossary  and  other  Addition, 
and  Improvements. 

FROM  THE  SKCOND  ENGLISH  EDITION, 
TRANSLATED    FROM    THE    SIXTH    GERMAN    EDITION, 

BY  HENRY  MEDLOCK,  F.  C.  S.,  &c. 

Illustrated  by  six  hundred  and  seventy-nine  Engravings  on  Wood. 
In  one  handsome  volume,  crown  octavo,  of  about  seven  hundred  large  pages. 

To  accommodate  those  who  desire  to  use  the  separate  portions  of  this  work, 
the  publishers  have  prepared  an  edition,  in  parts,  as  follows,  which  may  be  had 
singly,  neatly  done  up  in  flexible  cloth,  at  very  reasonable  prices. 

NATURAL  PHILOSOPHY,  .         .114  pages,  with  149  Illustrations 
ASTRONOMY,       .         .         .         .       64          «  5| 

CHEMISTRY,        ....     HO          »  48 

MINERALOGY  AND  GEOLOGY,     104          «  167 

BOTANY, 98          "  176 

ZOOLOGY  AND  PHYSIOLOGY,  .    106          «  84 

INTRODUCTION,  GLOSSARY,  INDEX,  &c.,  96  pages. 
Copies  may  also  be  had  beautifully  done  up  in  fancy  cloth,  with  gilt  stamps, 
suitable  for  holiday  presents  and  school  prizes. 

The  necessity  of  some  acquaintance  with  the  Natural  Sciences  is  now  sn  uni- 
versally admitted  in  all  thorough  education,  while  the  circle  of  facts  and  princi- 
ples embraced  in  the  study  has  enlarged  so  rapidly,  that  a  compendious  Manual 
like  the  BOOK  OF  NATURE  cannot  fail  to  supply  a  want  frequently  felt  and  ex- 
pressed by  a  large  and  growing  class. 

The  reputation  of  the  present  volume  in  England  and  Germany,  where  re- 
peated editions  have  been  rapidly  called  for,  is  sufficient  proof  of  the  author's 
success  in  condensing  and  popularizing  the  principles  of  his  numerous  subjects. 
The  publishers  therefore  would  merely  state  that,  in  reproducing  the  work, 
they  have  spared  no  pains  to  render  it  even  better  adapted  to  the  American 
student.  It  has  been  passed  through  the  press  under  the  care  of  a  competent 
editor,  who  has  corrected  such  errors  as  had  escaped  the  attention  of  the  English 
translator,  and  has  made  whatever  additions  appeared  necessary  to  bring  it  com- 
pletely on  a  level  with  the  existing  state  of  science.  These  will  be  found  prin- 
cipally in  the  sections  on  Botany  and  Geology;  especially  the  latter,  in  which 
references  have  been  made  to  the  numerous  and  systematic  Government  surveys 
of  the  several  States,  and  the  whole  adapted  to  the  nomenclature  and  systems 
generally  used  in  this  country.  A  copious  Glossary  has  been  appended,  and 
numerous  additional  illustrations  have  been  introduced  wherever  the  elucidation 
of  the  text  appeared  to  render  them  desirable. 

It  is  therefore  confidently  presented  as  an  excellent  Manual  for  the  private  stu- 
dent, or  as  a  complete  and  thorough  Class-book  for  collegiate  and  academical  use. 
Written  wiili  remarkable  clearness,  an<l  scrupulously  correct  in  its  details.— Mining 
Journal. 

His  exposivons  are  most  lurid.  There  are  few  who  will  not  follow  him  with  pleasure 
as  well  a*  with  profit  through  his  masterly  exposition  of  the  principles  and  primary  law* 
of  science.  It  should  certainly  be  made  a  class-book  in  school*.—  Critic. 


BLANCHARD  &  LEA'S  PUBLICATIONS.— (Educational  Works.} 


NEW  AND  IMPROVED  EDITION— (Now  Ready.) 

OUTLINES  OF  ENGLISH  LITERATURE. 

BY  THOMAS  B.  SHAW, 

Professor  of  English  Literature  in  the  Imperial  Alexander  Lyceum,  St.  Petersburg. 
SECOND    AMERICAN   EDITION. 

WITH  A  SKETCH  OF  AMERICAN  LITERATURE. 

BY  HENRY  T.  TUCKERMAN, 
Author  of  "Characteristics  of  Literature,'"-  The  Optimist."  &c. 

In  one  large  and  handsome  volume,  royal  12mo.,  extra  cloth,  of  about  500  pages. 

The  object  of  this  work  is  to  present  to  the  student  a  history  of  the  progress 
of  English  Literature.  To  accomplish  this,  the  author  hns  followed  its  course 
from  the  earliest  times  to  the  present  age,  seizing  upon  the  more  prominent 
"  Schools  of  Writing,"  tracing  their  causes  and  effects,  and  selecting  the  more 
celebrated  authors  as  subjects  for  brief  biographical  and  critical  sketches,  ana- 
lyzing their  best  works,  and  thus  presenting  to  the  student  a  definite  view  of  the 
development  of  the  language  and  literature,  with  succinct  descriptions  of  those 
books  and  men  of  which  no  educated  person  should  be  ignorant.  He  has  thus 
not  only  supplied  the  acknowledged  want  of  a  manual  on  this  subject,  but  by 
the  liveliness  and  power  of  his  style,  the  thorough  knowledge  he  displays  of  his 
topic,  and  the  variety  of  his  subjects,  he  has  succeeded  in  producing  a  most 
agreeable  reading-book,  which  will  captivate  the  mind  of  the  scholar,  and  re- 
lieve the  monotony  of  drier  studies. 

This  work  having  attracted  much  attention,  and  been  introduced  into  a  large 
number  of  our  best  academies  and  colleges,  the  publishers,  in  answering  the  call 
for  a  new  edition,  have  endeavored  to  render  it  still  more  appropriate  for  the 
student  of  this  country,  by  adding  to  it  a  sketch  of  American  literature.  This 
has  been  prepared  by  Mr.  Tuckerman,  on  the  plan  adopted  by  Mr.  Shaw,  and 
the  volume  is  again  presented  with  full  confidence  that  it  will  be  found  of  great 
utility  as  a  text-book,  wherever  this  subject  forms  part  of  the  educational  course ; 
or  as  an  introduction  to  a  systematic  plan  of  reading. 

From  Prof.  R.  P.  Dunn,  Brown  University,  April  22, 1?52. 

I  had  already  determined  to  adopt  it  as  the  principal  book  of  reference  in  my  depart- 
ment. This  is  the  first  term  iu  which  it  has  been  used  here  ;  but  from  the  irial  which  I 
have  now  made  of  it,  I  have  every  reason  lo  congratulate  mysrlf  on  my  selection  of  it 
as  a  text-book. 

From  the  Rev.  IF.  G.  T.  Shedd,  Professor  of  English  Literature  in  the  University  of  Vt. 
I  take  great  pleasure  in  saying  that  it  supplies  a  want  that  lias  long  existed  of  a  brief 
history  01  English  literature,  written  in  the  right  method  and  spirit,  to  serve  as  an  intro- 
duction to  the  critical  study  of  it.    I  shall  recommend  the  book  to  my  classes. 

From  James  Shannon,  President  of  Bacon  College,  Ky. 

I  have  read  about  one-half  of  "  Shaw's  Outlines."  and  so  far  I  am  more  than  pleased 
with  the  work.  1  concur  with  you  fully  in  the  opinion  that  it  supplies  a  want  long  felt 
in  our  higher  educational  institutes  of  n  criiical  history  of  English  literaiure.  occupying 
a  reasonable  space,  and  written  in  a  manner  10  interest  ana  attract  Hie  attention  of  the 
student.  I  sincerely  desire  that  it  may  obtam,  as  il  deserves,  an  extensive  circulaiioi:. 


HANDBOOK  OF  MODERN  EUROPEAN  LITERATURE, 

Jritish,  Danish,  Dutch,  French,  German,  Hungarian,  Italian,  Polish  and  Rus- 
sian, Portuguese,  Spanish,  and  Swedish.  With  a  full  Biographical  and 
Chronological  Index.  By  Mrs.  FOSTER.  In  one  large  royal  12mo.  volume, 
extra  cloth.  Uniform  with  "  Shaw's  Outlines  of  English  Literature." 


OC 


^  o 


v 


-v-  Uu 


-2. 


~  O 


=  H 


-  6 


tn. 


University  of  California 

SOUTHERN  REGIONAL  LIBRARY  FACILITY 

405  Hilgard  Avenue,  Los  Angeles,  CA  90024-1388 

Return  this  material  to  the  library 

from  which  it  was  borrowed. 


oc 


IHAPR171995 


4(5  MOV  01 2000 

REC'DYRL  JUL?,lt)0 


I'oi 


.     f      I     k   x   1+00  *  ' 
,  U,    '.  .   k  •> 


P  ;ur  ','.  "^  l 


•^   «•' 


f..far'.^-.fc 

i;..r';H*. 

s^-ffirrr-^4 ; ' 

_  J     .    .   (K^C  4    '.    I 
'**-&        '• 


'  ^ 

IS 

>  V 
— "i 


Cfl  w 

=   ^, 4 

^/« 


w^>/. 


